mmcdara

4924 Reputation

17 Badges

7 years, 6 days

MaplePrimes Activity


These are answers submitted by mmcdara


to @C_R


 

NULL

restart

NULL

assume(-1 <= a, a < 1)

solve(cos(x) > a, x)

Warning, solutions may have been lost

 

restart

 

restart

assume(-1 <= a, a < 1)

solve(cos(x) > a, x) assuming -1 <= a, a < 1

solve(cos(x) > a, x) assuming a > -1

solve(cos(x) > -1/2, x)

RealRange(Open(-(8/3)*Pi), Open(-(4/3)*Pi)), RealRange(Open(-(2/3)*Pi), Open((2/3)*Pi)), RealRange(Open((4/3)*Pi), Open((8/3)*Pi))

(1)

# Given a value of a, f(a) returns the domain D, not necessarily connected, such
# that any x ibn D verifies cos(x) < a

f := proc(a) solve(cos(x) > a, x) end proc

proc (a) solve(a < cos(x), x) end proc

(2)

f(-1);
f(0);
f(1);

[[-Pi, Pi], [Pi, 3*Pi]]

 

[[-(1/2)*Pi, (1/2)*Pi]]

 

[[-2*Pi-arccos(9/10), -2*Pi+arccos(9/10)], [-arccos(9/10), arccos(9/10)], [2*Pi-arccos(9/10), 2*Pi+arccos(9/10)]]

 

[]

(3)

# Procedure g belongs basicallly how procedure f does.
# The only difference is the form the domain D is returned.
#
# Procedure g is most aimed at ploting the domains D(a) (vertical axis)
# versus a.

g := proc(a)
  local s := solve(cos(x) > a, x):
  map(u -> [op(op(1, u)), op(op(2, u))], [s])
end proc

proc (a) local s; s := solve(a < cos(x), x); map(proc (u) options operator, arrow; [op(op(1, u)), op(op(2, u))] end proc, [s]) end proc

(4)

plots:-display(
  seq(seq(plot([[a, g(a)[k][1]], [a, g(a)[k][2]]], color=red, thickness=7), k=1..nops(g(a))), a in [seq](-1..1, 0.01))
  , labels=["a", "Domain"]
  , labeldirections=[default, "vertical"]
)

 

 

NULL

 

Download Domains.mw



what you want to achieve in this worksheet.
Because this is not a matter of solve or dsolve: for what I understand the last two commands of your worksheet are meaningless.

See corrections, developments, advices and comments in the attached file
Sam_mcdara.mw


in such a way that the interval (0, +oo) is mapped onto some finite interval, for instance (0, 1).


 

restart:

with(plots):

eq1 := diff(f(x), x, x, x)+(1/2)*cos(alpha)*x*(diff(f(x), x, x))+(1/2)*sin(alpha)*f(x)*(diff(f(x), x, x)) = 0;

eq2 := diff(g(x), x, x)+diff(g(x), x)+(diff(g(x), x))*(diff(h(x), x))+cos(alpha)*x*(diff(g(x), x))+sin(alpha)*f(x)*g(x) = 0;

eq3 := diff(g(x), x, x)+diff(h(x), x, x)+1/2*(cos(alpha)*x+sin(alpha)*f(x)) = 0;

ics := f(0)=0, D(f)(0)=1, (D@@2)(f)(0)=a[1], g(0)=1, D(g)(0)=a[2], h(0)=1, D(h)(0)=a[3];

#bcs:=f(x) , g(x), h(x) tends to 0 as x tends to infinity

diff(diff(diff(f(x), x), x), x)+(1/2)*cos(alpha)*x*(diff(diff(f(x), x), x))+(1/2)*sin(alpha)*f(x)*(diff(diff(f(x), x), x)) = 0

 

diff(diff(g(x), x), x)+diff(g(x), x)+(diff(g(x), x))*(diff(h(x), x))+cos(alpha)*x*(diff(g(x), x))+sin(alpha)*f(x)*g(x) = 0

 

diff(diff(g(x), x), x)+diff(diff(h(x), x), x)+(1/2)*cos(alpha)*x+(1/2)*sin(alpha)*f(x) = 0

 

f(0) = 0, (D(f))(0) = 1, ((D@@2)(f))(0) = a[1], g(0) = 1, (D(g))(0) = a[2], h(0) = 1, (D(h))(0) = a[3]

(1)

 

ICs ONLY

 

sys := {eq1, eq2, eq3, ics}:

# dsolve(sys); # It's unlikely that a formal solution does exist

Warning,  computation interrupted

 

params := convert(indets(sys, name) minus {x}, list)

[alpha, a[1], a[2], a[3]]

(2)

sol := dsolve(sys, numeric, parameters=params);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [alpha = alpha, a[1] = `a[1]`, a[2] = `a[2]`, a[3] = `a[3]`]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 24, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..54, {(1) = 7, (2) = 7, (3) = 0, (4) = 0, (5) = 4, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..11, {(1) = 0., (2) = 1., (3) = `a[1]`, (4) = 1., (5) = `a[2]`, (6) = 1., (7) = `a[3]`, (8) = Float(undefined), (9) = Float(undefined), (10) = Float(undefined), (11) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..7, {(1) = .1, (2) = .1, (3) = .1, (4) = .1, (5) = .1, (6) = .1, (7) = .1}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0}, datatype = float[8], order = C_order), Array(1..7, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 0}, datatype = integer[8]), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(x), Y[2] = diff(f(x),x), Y[3] = diff(diff(f(x),x),x), Y[4] = g(x), Y[5] = diff(g(x),x), Y[6] = h(x), Y[7] = diff(h(x),x)]`; YP[3] := -(1/2)*cos(Y[8])*X*Y[3]-(1/2)*sin(Y[8])*Y[1]*Y[3]; YP[5] := -cos(Y[8])*X*Y[5]-sin(Y[8])*Y[1]*Y[4]-Y[5]*Y[7]-Y[5]; YP[7] := cos(Y[8])*X*Y[5]+sin(Y[8])*Y[1]*Y[4]+Y[5]*Y[7]+Y[5]-(1/2)*cos(Y[8])*X-(1/2)*sin(Y[8])*Y[1]; YP[1] := Y[2]; YP[2] := Y[3]; YP[4] := Y[5]; YP[6] := Y[7]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(x), Y[2] = diff(f(x),x), Y[3] = diff(diff(f(x),x),x), Y[4] = g(x), Y[5] = diff(g(x),x), Y[6] = h(x), Y[7] = diff(h(x),x)]`; YP[3] := -(1/2)*cos(Y[8])*X*Y[3]-(1/2)*sin(Y[8])*Y[1]*Y[3]; YP[5] := -cos(Y[8])*X*Y[5]-sin(Y[8])*Y[1]*Y[4]-Y[5]*Y[7]-Y[5]; YP[7] := cos(Y[8])*X*Y[5]+sin(Y[8])*Y[1]*Y[4]+Y[5]*Y[7]+Y[5]-(1/2)*cos(Y[8])*X-(1/2)*sin(Y[8])*Y[1]; YP[1] := Y[2]; YP[2] := Y[3]; YP[4] := Y[5]; YP[6] := Y[7]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..11, {(1) = 0., (2) = 0., (3) = 1., (4) = `a[1]`, (5) = 1., (6) = `a[2]`, (7) = 1., (8) = `a[3]`, (9) = undefined, (10) = undefined, (11) = undefined}); _vmap := array( 1 .. 7, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3), ( 4 ) = (4), ( 5 ) = (5), ( 6 ) = (6), ( 7 ) = (7)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; _dat[4][26] := _EnvDSNumericSaveDigits; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, f(x), diff(f(x), x), diff(diff(f(x), x), x), g(x), diff(g(x), x), h(x), diff(h(x), x)], (4) = [alpha = alpha, a[1] = `a[1]`, a[2] = `a[2]`, a[3] = `a[3]`]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(3)

# example of use:
#
# Step 1: instanciate parameters

data := [$1..4]:

sol(parameters=data);

# Step 2 (illustration): plot some quantity

display(
  odeplot(sol, [x, f(x)], x=0..1, color=red, legend=typeset('f(x)'))
  , odeplot(sol, [x, g(x)], x=0..1, color=green, legend=typeset('g(x)'))
  , odeplot(sol, [x, h(x)], x=0..1, color=blue, legend=typeset('h(x)'))
)

[alpha = 1., `a[1]` = 2., `a[2]` = 3., `a[3]` = 4.]

 

 

 

ICs & BCs

What ICs do you want to replace by the BCs ?

 

 

A NAIVE APPROACH WHICH DOESN'T ALWAYS WORK

 

# Method 1
#
# Assuming you want to replace the ICs which contain the higher derivative order,
# and assuming that L is a reasonnable approximation of +oo

L := 10:
`ICs+BCs` :=  f(0)=0, D(f)(0)=1, g(0)=1, h(0)=1, f(L)=0, g(L)=0, h(L)=0;

# REMARK:
#
# As soon as your differential susyemen containsBCs, the use of option "parameters"
# indsolve/numerics is forbidden.
# Thus you have to instanciate the parameters BEFORE using dsolve.
SYS := {eq1, eq2, eq3, `ICs+BCs`}:

SOL := dsolve(eval(SYS, params=~data), numeric)

f(0) = 0, (D(f))(0) = 1, g(0) = 1, h(0) = 1, f(10) = 0, g(10) = 0, h(10) = 0

 

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(69, {(1) = .0, (2) = 0.870974538799821e-1, (3) = .1745417052888219, (4) = .2625088923065745, (5) = .351335413082271, (6) = .441141628480368, (7) = .5322524209294525, (8) = .6247531116215557, (9) = .7187297300229558, (10) = .8142599291152003, (11) = .9114381145697731, (12) = 1.0103563665629145, (13) = 1.111098198840413, (14) = 1.213763281598792, (15) = 1.3186791675912501, (16) = 1.4261650858855384, (17) = 1.5366301485334277, (18) = 1.650618521073082, (19) = 1.7685492694112557, (20) = 1.8909878333980874, (21) = 2.018378929398229, (22) = 2.1509822835922745, (23) = 2.2892654755116633, (24) = 2.433050263142334, (25) = 2.582755730966836, (26) = 2.737947557709467, (27) = 2.898962326759001, (28) = 3.0657001840761797, (29) = 3.2385497967905934, (30) = 3.417583274751644, (31) = 3.603222120179413, (32) = 3.795478991417435, (33) = 3.9946293320331607, (34) = 4.200706411297825, (35) = 4.413773715495762, (36) = 4.633547094846196, (37) = 4.859226077424993, (38) = 5.089866260044136, (39) = 5.322913436416088, (40) = 5.557195404172183, (41) = 5.789447512443844, (42) = 6.018744469465908, (43) = 6.242048401243784, (44) = 6.459054343127049, (45) = 6.667721976965457, (46) = 6.868384247053476, (47) = 7.060442330280947, (48) = 7.244522629091641, (49) = 7.420889208107754, (50) = 7.590189512801662, (51) = 7.752915270998523, (52) = 7.909612635317981, (53) = 8.060839800790653, (54) = 8.207075164579217, (55) = 8.34879205050435, (56) = 8.486398080361754, (57) = 8.62032368846796, (58) = 8.751045492322062, (59) = 8.878856985542582, (60) = 9.004163662219183, (61) = 9.127197349548947, (62) = 9.248455605267898, (63) = 9.368098822898368, (64) = 9.486466234547263, (65) = 9.603256248351462, (66) = 9.712926184486998, (67) = 9.815087396790323, (68) = 9.910482123703622, (69) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(69, 7, {(1, 1) = .0, (1, 2) = 1.0, (1, 3) = -.6480404462581693, (1, 4) = 1.0, (1, 5) = 6.712036177861089, (1, 6) = 1.0, (1, 7) = -1.0339022965224494, (2, 1) = 0.8464051294174721e-1, (2, 2) = .9436061686054273, (2, 3) = -.6463638669400693, (2, 4) = 1.5850713378368448, (2, 5) = 6.717417773533324, (2, 6) = .9094042394199642, (2, 7) = -1.0418743969366608, (3, 1) = .1646875320531142, (3, 2) = .887278670068506, (3, 3) = -.6414111238178749, (3, 4) = 2.171119032375489, (3, 5) = 6.675933519847483, (3, 6) = .8193503902119806, (3, 7) = -1.0080821198421106, (4, 1) = .24026671582228112, (4, 2) = .8311895326537709, (4, 3) = -.6333153088662633, (4, 4) = 2.7535181932393353, (4, 5) = 6.546026981489801, (4, 6) = .7350133142547782, (4, 7) = -.8908777952240621, (5, 1) = .3116135903367043, (5, 2) = .7754067791223028, (5, 3) = -.6222085552352093, (5, 4) = 3.323708015046929, (5, 5) = 6.259325797597795, (5, 6) = .666400426642658, (5, 7) = -.6218696933834122, (6, 1) = .3787587100797661, (6, 2) = .7201355307695683, (6, 3) = -.6082607888178513, (6, 4) = 3.863891404986298, (6, 5) = 5.721420277610916, (6, 6) = .6315129002256871, (6, 7) = -.10663578876260346, (7, 1) = .4418683876562277, (7, 2) = .6654563419813174, (7, 3) = -.5916289255947108, (7, 4) = 4.347933225749276, (7, 5) = 4.846656411734138, (7, 6) = .657810955795531, (7, 7) = .7404039816146503, (8, 1) = .500918960543993, (8, 2) = .6115978373830497, (8, 3) = -.5725278099484574, (8, 4) = 4.743008996404285, (8, 5) = 3.6582099187253725, (8, 6) = .7780598044376896, (8, 7) = 1.8960321617215627, (9, 1) = .5558974309344233, (9, 2) = .5587820368033454, (9, 3) = -.5512001365237005, (9, 4) = 5.025572356322828, (9, 5) = 2.367031182548753, (9, 6) = 1.0157172073378877, (9, 7) = 3.149247554704324, (10, 1) = .606797728294307, (10, 2) = .5072266502359998, (10, 3) = -.5279143275196628, (10, 4) = 5.196395978632939, (10, 5) = 1.2625890141348572, (10, 6) = 1.3698389725730888, (10, 7) = 4.210525767789734, (11, 1) = .6536352393722387, (11, 2) = .45712885926024804, (11, 3) = -.5029536377841997, (11, 4) = 5.278929795655956, (11, 5) = .496271373633428, (11, 6) = 1.8168552235916664, (11, 7) = 4.928407390684238, (12, 1) = .6964356133154653, (12, 2) = .40867405388264166, (12, 3) = -.4766152309685926, (12, 4) = 5.303168320490278, (12, 5) = 0.3857815921333268e-1, (12, 6) = 2.3265925807629206, (12, 7) = 5.332312132360375, (13, 1) = .7352338058629547, (13, 2) = .3620361291061937, (13, 3) = -.4492068847677641, (13, 4) = 5.293081438310534, (13, 5) = -.21195192764738605, (13, 6) = 2.874806346806758, (13, 7) = 5.523616246994811, (14, 1) = .7700843818909571, (14, 2) = .31736381113190426, (14, 3) = -.421034983641865, (14, 4) = 5.2637771210666084, (14, 5) = -.3444771677783103, (14, 6) = 3.4461456724458546, (14, 7) = 5.591373864414014, (15, 1) = .8011164027809063, (15, 2) = .2746980138301477, (15, 3) = -.3923416127616534, (15, 4) = 5.223530447232472, (15, 5) = -.4153410757063224, (15, 6) = 4.033210495417085, (15, 7) = 5.591654766990816, (16, 1) = .8284322275463121, (16, 2) = .2340890698620308, (16, 3) = -.3633687602618931, (16, 4) = 5.17655339911163, (16, 5) = -.4550032211877211, (16, 6) = 4.632484516117227, (16, 7) = 5.554602178166578, (17, 1) = .8521333774268491, (17, 2) = .19556132873239343, (17, 3) = -.33433382864743033, (17, 4) = 5.124867684147341, (17, 5) = -.47880505652617594, (17, 6) = 5.242938566820298, (17, 7) = 5.495125807185852, (18, 1) = .872316281452212, (18, 2) = .15911014322080141, (18, 3) = -.3054230190522077, (18, 4) = 5.069350266590356, (18, 5) = -.4941616843373842, (18, 6) = 5.865146077237633, (18, 7) = 5.420040217602523, (19, 1) = .8890234511880962, (19, 2) = .12478994286934476, (19, 3) = -.27686178417698876, (19, 4) = 5.010418556350278, (19, 5) = -.5045513814329267, (19, 6) = 6.499246214037362, (19, 7) = 5.332250558870125, (20, 1) = .9022983896210743, (20, 2) = 0.9262421869148249e-1, (20, 3) = -.24884721060056614, (20, 4) = 4.948178090393403, (20, 5) = -.5116089531163781, (20, 6) = 7.146100699458152, (20, 7) = 5.232628819067394, (21, 1) = .912153700163305, (21, 2) = 0.6267933747182403e-1, (21, 3) = -.2216116355974542, (21, 4) = 4.882691057764401, (21, 5) = -.5160972165963251, (21, 6) = 7.805672653378075, (21, 7) = 5.121204447168881, (22, 1) = .9185952042309146, (22, 2) = 0.3505394949569717e-1, (22, 3) = -.19542182485318538, (22, 4) = 4.814078723661528, (22, 5) = -.5183943208547083, (22, 6) = 8.476655611699533, (22, 7) = 4.9977355173050695, (23, 1) = .9216555193820289, (23, 2) = 0.978223046951856e-2, (23, 3) = -.17048921215045212, (23, 4) = 4.742348850895393, (23, 5) = -.5187241421028137, (23, 6) = 9.158427632916988, (23, 7) = 4.86157693484893, (24, 1) = .9213824652838725, (24, 2) = -0.13020136804823973e-1, (24, 3) = -.14711166715576876, (24, 4) = 4.66784998399372, (24, 5) = -.5172543316356903, (24, 6) = 9.846826687599357, (24, 7) = 4.712627040418187, (25, 1) = .9178682053535855, (25, 2) = -0.33388172195598954e-1, (25, 3) = -.12543963703164604, (25, 4) = 4.590629207508073, (25, 5) = -.5141417642145678, (25, 6) = 10.540261387365216, (25, 7) = 4.550147197120136, (26, 1) = .9112580107270338, (26, 2) = -0.5128861863125062e-1, (26, 3) = -.10569499178397904, (26, 4) = 4.511178024132107, (26, 5) = -.509566020920224, (26, 6) = 11.232856552646142, (26, 7) = 4.374304301123723, (27, 1) = .9017093137584035, (27, 2) = -0.6684121517057322e-1, (27, 3) = -0.8792954867072943e-1, (27, 4) = 4.4295893909536135, (27, 5) = -.5037001825635865, (27, 6) = 11.92199532035127, (27, 7) = 4.184416862985837, (28, 1) = .8894179650397273, (28, 2) = -0.8015484616341235e-1, (28, 3) = -0.7219418381711061e-1, (28, 4) = 4.34617348317203, (28, 5) = -.4967362184171179, (28, 6) = 12.60278002396964, (28, 7) = 3.9802764743904784, (29, 1) = .874556305758353, (29, 2) = -0.9140967105626117e-1, (29, 3) = -0.58440899474496245e-1, (29, 4) = 4.260986557019704, (29, 5) = -.48884907874152483, (29, 6) = 13.27193089748563, (29, 7) = 3.7610458267932096, (30, 1) = .857320587278922, (30, 2) = -.10077932104924725, (30, 3) = -0.4660987378994365e-1, (30, 4) = 4.174233015928716, (30, 5) = -.48022576342956846, (30, 6) = 13.924378009422584, (30, 7) = 3.526218935890464, (31, 1) = .837869536761922, (31, 2) = -.108468572639783, (31, 3) = -0.36581886526880374e-1, (31, 4) = 4.085935331485943, (31, 5) = -.4710341003293811, (31, 6) = 14.555764505512006, (31, 7) = 3.2747680844634757, (32, 1) = .8163939929224967, (32, 2) = -.11466879573903958, (32, 3) = -0.28233259001935695e-1, (32, 4) = 3.9962968936134, (32, 5) = -.46145819571300545, (32, 6) = 15.1596698960361, (32, 7) = 3.0061396458154253, (33, 1) = .7930457224151988, (33, 2) = -.11958351120350143, (33, 3) = -0.21402549872086916e-1, (33, 4) = 3.9053759811224826, (33, 5) = -.45166517321661986, (33, 6) = 15.729928940348364, (33, 7) = 2.7193560109166883, (34, 1) = .7679892122204723, (34, 2) = -.12340454034449624, (34, 3) = -0.15922054775886898e-1, (34, 4) = 3.813317990548415, (34, 5) = -.44183001829492036, (34, 6) = 16.258986445742764, (34, 7) = 2.413716029088757, (35, 1) = .7413692633527446, (35, 2) = -.12631623162897804, (35, 3) = -0.11612952204800629e-1, (35, 4) = 3.720222067461873, (35, 5) = -.4321226529677114, (35, 6) = 16.738781851120308, (35, 7) = 2.088424467428589, (36, 1) = .7133565766973178, (36, 2) = -.12848597328351313, (36, 3) = -0.8300218102609624e-2, (36, 4) = 3.626298067175515, (36, 5) = -.42272600574042085, (36, 6) = 17.16000416553925, (36, 7) = 1.7431885674897278, (37, 1) = .6841714392692512, (37, 2) = -.13006365696144934, (37, 3) = -0.5815667109713515e-2, (37, 4) = 3.5319158851001466, (37, 5) = -.4138402307384678, (37, 6) = 17.51245366265956, (37, 7) = 1.3785765022188865, (38, 1) = .6540363697931747, (38, 2) = -.13118330897340305, (38, 3) = -0.3997511270493869e-2, (38, 4) = 3.437429133858967, (38, 5) = -.4056642566037722, (38, 6) = 17.786435759356987, (38, 7) = .9955170115805172, (39, 1) = .6233686741962641, (39, 2) = -.1319553363936919, (39, 3) = -0.27054310603028806e-2, (39, 4) = 3.3437534209737163, (39, 5) = -.3984391577568871, (39, 6) = 17.97230947466633, (39, 7) = .5978811840872796, (40, 1) = .5923886781227782, (40, 2) = -.13247722265208312, (40, 3) = -0.18058506629828137e-2, (40, 4) = 3.251144552694234, (40, 5) = -.39234263763419475, (40, 6) = 18.064522047496506, (40, 7) = .18755516970091451, (41, 1) = .5615778766712999, (41, 2) = -.13282124913377782, (41, 3) = -0.11956476539588014e-2, (41, 4) = 3.1606004869508553, (41, 5) = -.38758928716647956, (41, 6) = 18.059843703274566, (41, 7) = -.2295419345534936, (42, 1) = .5310948728312335, (42, 2) = -.13304548731755467, (42, 3) = -0.7868264470365085e-3, (42, 4) = 3.0721310264580053, (42, 5) = -.3843208004193983, (42, 6) = 17.959049061990193, (42, 7) = -.6512460093265886, (43, 1) = .5013681319215946, (43, 2) = -.13318919670996712, (43, 3) = -0.5178498651861844e-3, (43, 4) = 2.986522714951428, (43, 5) = -.38269767105703134, (43, 6) = 17.766903427125378, (43, 7) = -1.071191473354842, (44, 1) = .4724546140682466, (44, 2) = -.13328117405154002, (44, 3) = -0.3413580353100642e-3, (44, 4) = 2.903492918924793, (44, 5) = -.3828390527629625, (44, 6) = 17.489393969659655, (44, 7) = -1.4878035729647847, (45, 1) = .44463663024154215, (45, 2) = -.13333964751039237, (45, 3) = -0.22648625942192163e-3, (45, 4) = 2.823430513966099, (45, 5) = -.38486618748706, (45, 6) = 17.136473183247634, (45, 7) = -1.8960240433469588, (46, 1) = .4178763871575324, (46, 2) = -.13337707429917123, (46, 3) = -0.15131817103957587e-3, (46, 4) = 2.7458339712387616, (46, 5) = -.38891533024117714, (46, 6) = 16.7160617281365, (46, 7) = -2.2952744658865645, (47, 1) = .3922577827117572, (47, 2) = -.1334011091944272, (47, 3) = -0.1020340304984026e-3, (47, 4) = 2.670582981808903, (47, 5) = -.39513417204427287, (47, 6) = 16.238075760804488, (47, 7) = -2.683133947090106, (48, 1) = .3676997391332355, (48, 2) = -.13341670713790443, (48, 3) = -0.6942089755924876e-4, (48, 4) = 2.5970988757829483, (48, 5) = -.40373576454086685, (48, 6) = 15.709576241208882, (48, 7) = -3.0596508048177626, (49, 1) = .3441685341542746, (49, 2) = -.1334269188594998, (49, 3) = -0.4767489164919368e-4, (49, 4) = 2.5249500375664162, (49, 5) = -.4149783516600292, (49, 6) = 15.137859491202565, (49, 7) = -3.4241912786655817, (50, 1) = .3215787090342546, (50, 2) = -.1334336783301011, (50, 3) = -0.3302998128677364e-4, (50, 4) = 2.4535422175468073, (50, 5) = -.42921306049055047, (50, 6) = 14.528315573551494, (50, 7) = -3.776945775772297, (51, 1) = .2998652229621313, (51, 2) = -.13343819731938128, (51, 3) = -0.23078387473751626e-4, (51, 4) = 2.3823203484189173, (51, 5) = -.4468749509629552, (51, 6) = 13.885993756754347, (51, 7) = -4.117802985772745, (52, 1) = .2789555561020756, (52, 2) = -.13344124906669236, (52, 3) = -0.16253421185514632e-4, (52, 4) = 2.3106671066390487, (52, 5) = -.4685168932998098, (52, 6) = 13.214974102927906, (52, 7) = -4.446754194703429, (53, 1) = .25877544789038537, (53, 2) = -.13344333033339548, (53, 3) = -0.11530162618361092e-4, (53, 4) = 2.237899908851254, (53, 5) = -.4948362168297376, (53, 6) = 12.518519526075229, (53, 7) = -4.7637718541674765, (54, 1) = .23926120335249454, (54, 2) = -.13344476251372298, (54, 3) = -0.8234296560410107e-5, (54, 4) = 2.1632943062171615, (54, 5) = -.5266897716233722, (54, 6) = 11.799570745691826, (54, 7) = -5.068576798223696, (55, 1) = .22034975285929972, (55, 2) = -.13344575651242535, (55, 3) = -0.5916145796033301e-5, (55, 4) = 2.08602785697325, (55, 5) = -.5651371343720492, (55, 6) = 11.060516786018038, (55, 7) = -5.360752185836024, (56, 1) = .20198676168298482, (56, 2) = -.13344645174444336, (56, 3) = -0.4273957304694728e-5, (56, 4) = 2.005188110401573, (56, 5) = -.6114597795955431, (56, 6) = 10.303593940339832, (56, 7) = -5.639574562212925, (57, 1) = .1841148299286156, (57, 2) = -.13344694166642246, (57, 3) = -0.31024754988541686e-5, (57, 4) = 1.919695271175956, (57, 5) = -.6672215586738615, (57, 6) = 9.530505189523884, (57, 7) = -5.904152609600287, (58, 1) = .16667038106786794, (58, 2) = -.1334472894408278, (58, 3) = -0.22610334620838017e-5, (58, 4) = 1.828240308209183, (58, 5) = -.7343324080988083, (58, 6) = 8.742299330292639, (58, 7) = -6.153420108144532, (59, 1) = .1496142670499021, (59, 2) = -.13344753765933798, (59, 3) = -0.16535947547906961e-5, (59, 4) = 1.7294070592248962, (59, 5) = -.8149525184539348, (59, 6) = 7.9408416615750355, (59, 7) = -6.385670433272607, (60, 1) = .13289238785388657, (60, 2) = -.1334477158641132, (60, 3) = -0.12126630806542414e-5, (60, 4) = 1.621431554527366, (60, 5) = -.911628839350124, (60, 6) = 7.127135272301367, (60, 7) = -6.599126835468909, (61, 1) = .11647381499838148, (61, 2) = -.13344784432213253, (61, 3) = -0.8914039009218215e-6, (61, 4) = 1.502397673795262, (61, 5) = -1.0270410604815439, (61, 6) = 6.303192963834142, (61, 7) = -6.791491090936316, (62, 1) = .1002921562327314, (62, 2) = -.13344793744182815, (62, 3) = -0.6560866327634551e-6, (62, 4) = 1.3697944437952236, (62, 5) = -1.1642673373452328, (62, 6) = 5.469183300792872, (62, 7) = -6.9607696083858235, (63, 1) = 0.843260113579988e-1, (63, 2) = -.13344800509804405, (63, 3) = -0.4833718993502656e-6, (63, 4) = 1.2211010250134742, (63, 5) = -1.3259717665708373, (63, 6) = 4.6275258079102235, (63, 7) = -7.104571686201826, (64, 1) = 0.685301133376551e-1, (64, 2) = -.13344805441766813, (64, 3) = -0.356210919319984e-6, (64, 4) = 1.0532788664635098, (64, 5) = -1.514590088430831, (64, 6) = 3.7794093211844757, (64, 7) = -7.221216674253433, (65, 1) = 0.52944711018845826e-1, (65, 2) = -.13344809029739021, (65, 3) = -0.262798879514074e-6, (65, 4) = .8640745423175178, (65, 5) = -1.7305096507790398, (65, 6) = 2.9306271549805953, (65, 7) = -7.309431114359894, (66, 1) = 0.38309466039322865e-1, (66, 2) = -.13344811534197976, (66, 3) = -0.1969872464992203e-6, (66, 4) = .6619111576411697, (66, 5) = -1.9606156271817812, (66, 6) = 2.125598155483905, (66, 7) = -7.367575172853756, (67, 1) = 0.24676243855094707e-1, (67, 2) = -.13344813297478225, (67, 3) = -0.15025123783830536e-6, (67, 4) = .44967727275252567, (67, 5) = -2.197780877190878, (67, 6) = 1.3710376052586668, (67, 7) = -7.401240098998529, (68, 1) = 0.11945995023689765e-1, (68, 2) = -.13344814562899987, (68, 3) = -0.11644279198799223e-6, (68, 4) = .2287312307078018, (68, 5) = -2.437142166544135, (68, 6) = .6641446705800779, (68, 7) = -7.416787525049909, (69, 1) = .0, (69, 2) = -.13344815489318823, (69, 3) = -0.915077785296785e-7, (69, 4) = .0, (69, 5) = -2.675077658593984, (69, 6) = .0, (69, 7) = -7.419828149579798}, datatype = float[8], order = C_order); YP := Matrix(69, 7, {(1, 1) = 1.0, (1, 2) = -.6480404462581693, (1, 3) = .0, (1, 4) = 6.712036177861089, (1, 5) = .2275534407712545, (1, 6) = -1.0339022965224494, (1, 7) = -.2275534407712545, (2, 1) = .9436061686054273, (2, 2) = -.6463638669400693, (2, 3) = 0.382264409377552e-1, (2, 4) = 6.717417773533324, (2, 5) = -.1477196436742213, (2, 6) = -1.0418743969366608, (2, 7) = 0.8857889820105348e-1, (3, 1) = .887278670068506, (3, 2) = -.6414111238178749, (3, 3) = 0.7468753582986948e-1, (3, 4) = 6.675933519847483, (3, 5) = -.8764933214451714, (3, 6) = -1.0080821198421106, (3, 7) = .7600507886351592, (4, 1) = .8311895326537709, (4, 2) = -.6333153088662633, (4, 3) = .10893391579134806, (4, 4) = 6.546026981489801, (4, 5) = -2.1994664757004054, (4, 6) = -.8908777952240621, (4, 7) = 2.027460660798572, (5, 1) = .7754067791223028, (5, 2) = -.6222085552352093, (5, 3) = .14063192875461494, (5, 4) = 6.259325797597795, (5, 5) = -4.426554001970522, (5, 6) = -.6218696933834122, (5, 7) = 4.200533437689703, (6, 1) = .7201355307695683, (6, 2) = -.6082607888178513, (6, 3) = .16942018644339363, (6, 4) = 5.721420277610916, (6, 5) = -7.706489797047124, (6, 6) = -.10663578876260346, (6, 7) = 7.427957645118244, (7, 1) = .6654563419813174, (7, 2) = -.5916289255947108, (7, 3) = .19505906213641666, (7, 4) = 4.846656411734138, (7, 5) = -11.445574088833629, (7, 6) = .7404039816146503, (7, 7) = 11.115875770009305, (8, 1) = .6115978373830497, (8, 2) = -.5725278099484574, (8, 3) = .21729271575300135, (8, 4) = 3.6582099187253725, (8, 5) = -13.828362521505262, (8, 6) = 1.8960321617215627, (8, 7) = 13.4488303625826, (9, 1) = .5587820368033454, (9, 2) = -.5512001365237005, (9, 3) = .23594201467180825, (9, 4) = 2.367031182548753, (9, 5) = -13.091410528660168, (9, 6) = 3.149247554704324, (9, 7) = 12.66335908411619, (10, 1) = .5072266502359998, (10, 2) = -.5279143275196628, (10, 3) = .2509042706571769, (10, 4) = 1.2625890141348572, (10, 5) = -9.7875179552591, (10, 6) = 4.210525767789734, (10, 7) = 9.312243355617113, (11, 1) = .45712885926024804, (11, 2) = -.5029536377841997, (11, 3) = .2621563361450444, (11, 4) = .496271373633428, (11, 5) = -6.08997980593615, (11, 6) = 4.928407390684238, (11, 7) = 5.56874620416722, (12, 1) = .40867405388264166, (12, 2) = -.4766152309685926, (12, 3) = .26974711881916036, (12, 4) = 0.3857815921333268e-1, (12, 5) = -3.373166328130113, (12, 6) = 5.332312132360375, (12, 7) = 2.807202210132936, (13, 1) = .3620361291061937, (13, 2) = -.4492068847677641, (13, 3) = .2737931311077191, (13, 4) = -.21195192764738605, (13, 5) = -1.7647786761428932, (13, 6) = 5.523616246994811, (13, 7) = 1.155275259361408, (14, 1) = .31736381113190426, (14, 2) = -.421034983641865, (14, 3) = .2744732875045658, (14, 4) = -.3444771677783103, (14, 5) = -.914461239677371, (14, 6) = 5.591373864414014, (14, 7) = .2625598581569353, (15, 1) = .2746980138301477, (15, 2) = -.3923416127616534, (15, 3) = .2720107546568691, (15, 4) = -.4153410757063224, (15, 5) = -.48755710754779136, (15, 6) = 5.591654766990816, (15, 7) = -.20574369412402893, (16, 1) = .2340890698620308, (16, 2) = -.3633687602618931, (16, 3) = .26665125468211026, (16, 4) = -.4550032211877211, (16, 5) = -.2756115671319668, (16, 6) = 5.554602178166578, (16, 7) = -.4582194162743254, (17, 1) = .19556132873239343, (17, 2) = -.33433382864743033, (17, 3) = .2586557491614539, (17, 4) = -.47880505652617594, (17, 5) = -.16733888036025313, (17, 6) = 5.495125807185852, (17, 7) = -.6063062820448284, (18, 1) = .15911014322080141, (18, 2) = -.3054230190522077, (18, 3) = .24828781486886087, (18, 4) = -.4941616843373842, (18, 5) = -.10780171642104186, (18, 6) = 5.420040217602523, (18, 7) = -.7051292003099578, (19, 1) = .12478994286934476, (19, 2) = -.27686178417698876, (19, 3) = .23583622327842457, (19, 4) = -.5045513814329267, (19, 5) = -0.7116071666160817e-1, (19, 6) = 5.332250558870125, (19, 7) = -.7806586269848306, (20, 1) = 0.9262421869148249e-1, (20, 2) = -.24884721060056614, (20, 3) = .22159383752300865, (20, 4) = -.5116089531163781, (20, 5) = -0.4556120239783468e-1, (20, 6) = 5.232628819067394, (20, 7) = -.844920298231469, (21, 1) = 0.6267933747182403e-1, (21, 2) = -.2216116355974542, (21, 3) = .20588670132266562, (21, 4) = -.5160972165963251, (21, 5) = -0.25755233962067248e-1, (21, 6) = 5.121204447168881, (21, 7) = -.9032875970589895, (22, 1) = 0.3505394949569717e-1, (22, 2) = -.19542182485318538, (22, 3) = .18908545772072805, (22, 4) = -.5183943208547083, (22, 5) = -0.9484363097654658e-2, (22, 6) = 4.9977355173050695, (22, 7) = -.9580915863275563, (23, 1) = 0.978223046951856e-2, (23, 2) = -.17048921215045212, (23, 3) = .17154980786899687, (23, 4) = -.5187241421028137, (23, 5) = 0.4237507619642145e-2, (23, 6) = 4.86157693484893, (23, 7) = -1.010458403975294, (24, 1) = -0.13020136804823973e-1, (24, 2) = -.14711166715576876, (24, 3) = .1537242834809326, (24, 4) = -.5172543316356903, (24, 5) = 0.15793033051178185e-1, (24, 6) = 4.712627040418187, (24, 7) = -1.0607426720091961, (25, 1) = -0.33388172195598954e-1, (25, 2) = -.12543963703164604, (25, 3) = .13596580003431602, (25, 4) = -.5141417642145678, (25, 5) = 0.25415393785422236e-1, (25, 6) = 4.550147197120136, (25, 7) = -1.1093295635945584, (26, 1) = -0.5128861863125062e-1, (26, 2) = -.10569499178397904, (26, 3) = .11870163538565442, (26, 4) = -.509566020920224, (26, 5) = 0.3321517946442043e-1, (26, 6) = 4.374304301123723, (26, 7) = -1.1562734567029342, (27, 1) = -0.6684121517057322e-1, (27, 2) = -0.8792954867072943e-1, (27, 3) = .10222154075055834, (27, 4) = -.5037001825635865, (27, 5) = 0.3934029168764619e-1, (27, 6) = 4.184416862985837, (27, 7) = -1.2018794187033923, (28, 1) = -0.8015484616341235e-1, (28, 2) = -0.7219418381711061e-1, (28, 3) = 0.8680716340217215e-1, (28, 4) = -.4967362184171179, (28, 5) = 0.4391939994977978e-1, (28, 6) = 3.9802764743904784, (28, 7) = -1.2463315447020582, (29, 1) = -0.9140967105626117e-1, (29, 2) = -0.58440899474496245e-1, (29, 3) = 0.7263355472934034e-1, (29, 4) = -.48884907874152483, (29, 5) = 0.47100370241660805e-1, (29, 6) = 3.7610458267932096, (29, 7) = -1.2899552096172555, (30, 1) = -.10077932104924725, (30, 2) = -0.4660987378994365e-1, (30, 3) = 0.59845645222119034e-1, (30, 4) = -.48022576342956846, (30, 5) = 0.4902221689204245e-1, (30, 6) = 3.526218935890464, (30, 7) = -1.332991478251233, (31, 1) = -.108468572639783, (31, 2) = -0.36581886526880374e-1, (31, 3) = 0.4850524256797501e-1, (31, 4) = -.4710341003293811, (31, 5) = 0.4982477631511362e-1, (31, 6) = 3.2747680844634757, (31, 7) = -1.3757608384789215, (32, 1) = -.11466879573903958, (32, 2) = -0.28233259001935695e-1, (32, 3) = 0.3864678472237587e-1, (32, 4) = -.46145819571300545, (32, 5) = 0.4963758794431161e-1, (32, 6) = 3.0061396458154253, (32, 7) = -1.4184765420206453, (33, 1) = -.11958351120350143, (33, 2) = -0.21402549872086916e-1, (33, 3) = 0.30237869225799123e-1, (33, 4) = -.45166517321661986, (33, 5) = 0.4858099023524076e-1, (33, 6) = 2.7193560109166883, (33, 7) = -1.461397192347457, (34, 1) = -.12340454034449624, (34, 2) = -0.15922054775886898e-1, (34, 3) = 0.23213496064549432e-1, (34, 4) = -.44183001829492036, (34, 5) = 0.46761263741275094e-1, (34, 6) = 2.413716029088757, (34, 7) = -1.5047072632554226, (35, 1) = -.12631623162897804, (35, 2) = -0.11612952204800629e-1, (35, 3) = 0.1746943855800931e-1, (35, 4) = -.4321226529677114, (35, 5) = 0.4426619934595255e-1, (35, 6) = 2.088424467428589, (35, 7) = -1.5485726194470955, (36, 1) = -.12848597328351313, (36, 2) = -0.8300218102609624e-2, (36, 3) = 0.1288104639115637e-1, (36, 4) = -.42272600574042085, (36, 5) = 0.4116473023772382e-1, (36, 6) = 1.7431885674897278, (36, 7) = -1.593057250641075, (37, 1) = -.13006365696144934, (37, 2) = -0.5815667109713515e-2, (37, 3) = 0.9308444734499193e-2, (37, 4) = -.4138402307384678, (37, 5) = 0.3750715946256644e-1, (37, 6) = 1.3785765022188865, (37, 7) = -1.638087894035893, (38, 1) = -.13118330897340305, (38, 2) = -0.3997511270493869e-2, (38, 3) = 0.6596731284900002e-2, (38, 4) = -.4056642566037722, (38, 5) = 0.33315439953977455e-1, (38, 6) = .9955170115805172, (38, 7) = -1.683524992480146, (39, 1) = -.1319553363936919, (39, 2) = -0.27054310603028806e-2, (39, 3) = 0.4599948464520586e-2, (39, 4) = -.3984391577568871, (39, 5) = 0.28607777141917756e-1, (39, 6) = .5978811840872796, (39, 7) = -1.7288723050451378, (40, 1) = -.13247722265208312, (40, 2) = -0.18058506629828137e-2, (40, 3) = 0.3161180750301622e-2, (40, 4) = -.39234263763419475, (40, 5) = 0.2333933363340135e-1, (40, 6) = .18755516970091451, (40, 7) = -1.7738610213349426, (41, 1) = -.13282124913377782, (41, 2) = -0.11956476539588014e-2, (41, 3) = 0.2152526461939175e-2, (41, 4) = -.38758928716647956, (41, 5) = 0.17474209507355454e-1, (41, 6) = -.2295419345534936, (41, 7) = -1.8177758743098065, (42, 1) = -.13304548731755467, (42, 2) = -0.7868264470365085e-3, (42, 3) = 0.1455173527980766e-2, (42, 4) = -.3843208004193983, (42, 5) = 0.10883979419293865e-1, (42, 6) = -.6512460093265886, (42, 7) = -1.8603051998949274, (43, 1) = -.13318919670996712, (43, 2) = -0.5178498651861844e-3, (43, 3) = 0.982485447220071e-3, (43, 4) = -.38269767105703134, (43, 5) = 0.34644114209232413e-2, (43, 6) = -1.071191473354842, (43, 7) = -1.9007043515468791, (44, 1) = -.13328117405154002, (44, 2) = -0.3413580353100642e-3, (44, 3) = 0.6634974092188953e-3, (44, 4) = -.3828390527629625, (44, 5) = -0.500596637768469e-2, (44, 6) = -1.4878035729647847, (44, 7) = -1.9386934359704209, (45, 1) = -.13333964751039237, (45, 2) = -0.22648625942192163e-3, (45, 3) = 0.4503378474664687e-3, (45, 4) = -.38486618748706, (45, 5) = -0.14719192247822477e-1, (45, 6) = -1.8960240433469588, (45, 7) = -1.9736479988387396, (46, 1) = -.13337707429917123, (46, 2) = -0.15131817103957587e-3, (46, 3) = 0.30737522630137654e-3, (46, 4) = -.38891533024117714, (46, 5) = -0.26005757329816248e-1, (46, 6) = -2.2952744658865645, (46, 7) = -2.0053115933205916, (47, 1) = -.1334011091944272, (47, 2) = -0.1020340304984026e-3, (47, 3) = 0.2114577131265041e-3, (47, 4) = -.39513417204427287, (47, 5) = -0.3920524637781431e-1, (47, 6) = -2.683133947090106, (47, 7) = -2.0332181607306072, (48, 1) = -.13341670713790443, (48, 2) = -0.6942089755924876e-4, (48, 3) = 0.14660447261601823e-3, (48, 4) = -.40373576454086685, (48, 5) = -0.54804016848034154e-1, (48, 6) = -3.0596508048177626, (48, 7) = -2.0570164546590575, (49, 1) = -.1334269188594998, (49, 2) = -0.4767489164919368e-4, (49, 3) = 0.10248031150915673e-3, (49, 4) = -.4149783516600292, (49, 5) = -0.7336674171302004e-1, (49, 6) = -3.4241912786655817, (49, 7) = -2.0761989513406314, (50, 1) = -.1334336783301011, (50, 2) = -0.3302998128677364e-4, (50, 3) = 0.721968678413968e-4, (50, 4) = -.42921306049055047, (50, 5) = -0.95626412430874e-1, (50, 6) = -3.776945775772297, (50, 7) = -2.0901716119327354, (51, 1) = -.13343819731938128, (51, 2) = -0.23078387473751626e-4, (51, 3) = 0.5124839667238189e-4, (51, 4) = -.4468749509629552, (51, 5) = -.12247138504095234, (51, 6) = -4.117802985772745, (51, 7) = -2.098151556257248, (52, 1) = -.13344124906669236, (52, 2) = -0.16253421185514632e-4, (52, 3) = 0.3663777087506696e-4, (52, 4) = -.4685168932998098, (52, 5) = -.1550070679432004, (52, 6) = -4.446754194703429, (52, 7) = -2.099150408005264, (53, 1) = -.13344333033339548, (53, 2) = -0.11530162618361092e-4, (53, 3) = 0.26363961050608267e-4, (53, 4) = -.4948362168297376, (53, 5) = -.19460248472524794, (53, 6) = -4.7637718541674765, (53, 7) = -2.0919186965653993, (54, 1) = -.13344476251372298, (54, 2) = -0.8234296560410107e-5, (54, 3) = 0.19085588418241003e-4, (54, 4) = -.5266897716233722, (54, 5) = -.24291545443163842, (54, 6) = -5.068576798223696, (54, 7) = -2.0749010437016824, (55, 1) = -.13344575651242535, (55, 2) = -0.5916145796033301e-5, (55, 3) = 0.13891966749292008e-4, (55, 4) = -.5651371343720492, (55, 5) = -.301948900357193, (55, 6) = -5.360752185836024, (55, 7) = -2.046195859463701, (56, 1) = -.13344645174444336, (56, 2) = -0.4273957304694728e-5, (56, 3) = 0.10161731932876351e-4, (56, 4) = -.6114597795955431, (56, 5) = -.37404915356970014, (56, 6) = -5.639574562212925, (56, 7) = -2.003544071733275, (57, 1) = -.13344694166642246, (57, 2) = -0.31024754988541686e-5, (57, 3) = 0.7465344161959493e-5, (57, 4) = -.6672215586738615, (57, 5) = -.46193123039749057, (57, 6) = -5.904152609600287, (57, 7) = -1.9443227963358891, (58, 1) = -.1334472894408278, (58, 2) = -0.22610334620838017e-5, (58, 3) = 0.5503873616742709e-5, (58, 4) = -.7343324080988083, (58, 5) = -.5686530962592362, (58, 6) = -6.153420108144532, (58, 7) = -1.8655760777178079, (59, 1) = -.13344753765933798, (59, 2) = -0.16535947547906961e-5, (59, 3) = 0.4070458229934477e-5, (59, 4) = -.8149525184539348, (59, 5) = -.6972464819664995, (59, 6) = -6.385670433272607, (59, 7) = -1.7643350017324464, (60, 1) = -.1334477158641132, (60, 2) = -0.12126630806542414e-5, (60, 3) = 0.30175880678214197e-5, (60, 4) = -.911628839350124, (60, 5) = -.8505949161452631, (60, 6) = -6.599126835468909, (60, 7) = -1.6378028226507484, (61, 1) = -.13344784432213253, (61, 2) = -0.8914039009218215e-6, (61, 3) = 0.22416379522317367e-5, (61, 4) = -1.0270410604815439, (61, 5) = -1.0305508524242595, (61, 6) = -6.791491090936316, (61, 7) = -1.484176702518687, (62, 1) = -.13344793744182815, (62, 2) = -0.6560866327634551e-6, (62, 3) = 0.16669044897252967e-5, (62, 4) = -1.1642673373452328, (62, 5) = -1.23773082646301, (62, 6) = -6.9607696083858235, (62, 7) = -1.302946587896506, (63, 1) = -.13344800509804405, (63, 2) = -0.4833718993502656e-6, (63, 3) = 0.12404684323914335e-5, (63, 4) = -1.3259717665708373, (63, 5) = -1.4695906092434847, (63, 6) = -7.104571686201826, (63, 7) = -1.0966910344739813, (64, 1) = -.13344805441766813, (64, 2) = -0.356210919319984e-6, (64, 3) = 0.9231607927967096e-6, (64, 4) = -1.514590088430831, (64, 5) = -1.7202098601654023, (64, 6) = -7.221216674253433, (64, 7) = -.8714029813472833, (65, 1) = -.13344809029739021, (65, 2) = -0.262798879514074e-6, (65, 3) = 0.6876412475186891e-6, (65, 4) = -1.7305096507790398, (65, 5) = -1.9779984064061022, (65, 6) = -7.309431114359894, (65, 7) = -.6386080600680575, (66, 1) = -.13344811534197976, (66, 2) = -0.1969872464992203e-6, (66, 3) = 0.5200613725387069e-6, (66, 4) = -1.9606156271817812, (66, 5) = -2.2165580286922117, (66, 6) = -7.367575172853756, (66, 7) = -.42351833046825554, (67, 1) = -.13344813297478225, (67, 2) = -0.15025123783830536e-6, (67, 3) = 0.3999596820875845e-6, (67, 4) = -2.197780877190878, (67, 5) = -2.422777019011979, (67, 6) = -7.401240098998529, (67, 7) = -.23916232898863993, (68, 1) = -.13344814562899987, (68, 2) = -0.11644279198799223e-6, (68, 3) = 0.3123408209924974e-6, (68, 4) = -2.437142166544135, (68, 5) = -2.5908639424164606, (68, 6) = -7.416787525049909, (68, 7) = -0.9149033353311023e-1, (69, 1) = -.13344815489318823, (69, 2) = -0.915077785296785e-7, (69, 3) = 0.24720931872228184e-6, (69, 4) = -2.675077658593984, (69, 5) = -2.7200325818069295, (69, 6) = -7.419828149579798, (69, 7) = 0.1852105246623026e-1}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(69, {(1) = .0, (2) = 0.870974538799821e-1, (3) = .1745417052888219, (4) = .2625088923065745, (5) = .351335413082271, (6) = .441141628480368, (7) = .5322524209294525, (8) = .6247531116215557, (9) = .7187297300229558, (10) = .8142599291152003, (11) = .9114381145697731, (12) = 1.0103563665629145, (13) = 1.111098198840413, (14) = 1.213763281598792, (15) = 1.3186791675912501, (16) = 1.4261650858855384, (17) = 1.5366301485334277, (18) = 1.650618521073082, (19) = 1.7685492694112557, (20) = 1.8909878333980874, (21) = 2.018378929398229, (22) = 2.1509822835922745, (23) = 2.2892654755116633, (24) = 2.433050263142334, (25) = 2.582755730966836, (26) = 2.737947557709467, (27) = 2.898962326759001, (28) = 3.0657001840761797, (29) = 3.2385497967905934, (30) = 3.417583274751644, (31) = 3.603222120179413, (32) = 3.795478991417435, (33) = 3.9946293320331607, (34) = 4.200706411297825, (35) = 4.413773715495762, (36) = 4.633547094846196, (37) = 4.859226077424993, (38) = 5.089866260044136, (39) = 5.322913436416088, (40) = 5.557195404172183, (41) = 5.789447512443844, (42) = 6.018744469465908, (43) = 6.242048401243784, (44) = 6.459054343127049, (45) = 6.667721976965457, (46) = 6.868384247053476, (47) = 7.060442330280947, (48) = 7.244522629091641, (49) = 7.420889208107754, (50) = 7.590189512801662, (51) = 7.752915270998523, (52) = 7.909612635317981, (53) = 8.060839800790653, (54) = 8.207075164579217, (55) = 8.34879205050435, (56) = 8.486398080361754, (57) = 8.62032368846796, (58) = 8.751045492322062, (59) = 8.878856985542582, (60) = 9.004163662219183, (61) = 9.127197349548947, (62) = 9.248455605267898, (63) = 9.368098822898368, (64) = 9.486466234547263, (65) = 9.603256248351462, (66) = 9.712926184486998, (67) = 9.815087396790323, (68) = 9.910482123703622, (69) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(69, 7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = -0.8564770979105852e-15, (1, 4) = .0, (1, 5) = -0.13537476047190115e-12, (1, 6) = .0, (1, 7) = 0.13350842648359793e-12, (2, 1) = 0.22018148074602227e-15, (2, 2) = 0.7339773250778581e-15, (2, 3) = -0.19733430778675627e-14, (2, 4) = -0.6737763661199543e-13, (2, 5) = -0.645184239961593e-12, (2, 6) = 0.6782082361767538e-13, (2, 7) = 0.6365950339354017e-12, (3, 1) = -0.12654081931113597e-15, (3, 2) = 0.7695341896792135e-15, (3, 3) = 0.8587149998769551e-15, (3, 4) = -0.602554814783395e-13, (3, 5) = -0.6178864103501279e-12, (3, 6) = 0.6017766185038254e-13, (3, 7) = 0.6168801934609043e-12, (4, 1) = -0.4037391825813435e-16, (4, 2) = -0.10386148207273655e-14, (4, 3) = -0.17868609932547283e-14, (4, 4) = 0.21727136612089437e-12, (4, 5) = 0.11833425383023225e-11, (4, 6) = -0.21512120399469783e-12, (4, 7) = -0.11777482336877145e-11, (5, 1) = -0.43676960356980447e-15, (5, 2) = -0.5899812357056545e-15, (5, 3) = 0.9748850283953862e-15, (5, 4) = 0.3160057639100901e-12, (5, 5) = 0.15561487819677273e-11, (5, 6) = -0.31413227800132004e-12, (5, 7) = -0.15524261763369067e-11, (6, 1) = -0.4880452572324069e-15, (6, 2) = -0.25401447529753905e-15, (6, 3) = -0.8773026736409608e-15, (6, 4) = -0.39511606066974403e-12, (6, 5) = -0.2626518990564851e-11, (6, 6) = 0.3914519622748979e-12, (6, 7) = 0.26219980966942188e-11, (7, 1) = 0.3083402836537361e-15, (7, 2) = 0.11478877855242781e-14, (7, 3) = -0.4544718643563174e-15, (7, 4) = -0.364828543164158e-12, (7, 5) = -0.6979135076992772e-12, (7, 6) = 0.36611593792043347e-12, (7, 7) = 0.6843702801146759e-12, (8, 1) = -0.13902015859230462e-15, (8, 2) = -0.8141170235507663e-15, (8, 3) = -0.11062888629327639e-14, (8, 4) = -0.7638607130259561e-12, (8, 5) = 0.4222560971822865e-11, (8, 6) = 0.7682396744296017e-12, (8, 7) = -0.42191059007838435e-11, (9, 1) = -0.1192829525925201e-14, (9, 2) = -0.19715067920581095e-14, (9, 3) = 0.4828869923466516e-15, (9, 4) = 0.3030667045996736e-11, (9, 5) = -0.5663561193084528e-11, (9, 6) = -0.3025234671043208e-11, (9, 7) = 0.5656191257333786e-11, (10, 1) = -0.49322874886051877e-16, (10, 2) = 0.4667882345104275e-15, (10, 3) = 0.17418274038751516e-14, (10, 4) = -0.1013938045181613e-11, (10, 5) = -0.6511201510115692e-11, (10, 6) = 0.10089852826865726e-11, (10, 7) = 0.6511851401629007e-11, (11, 1) = -0.9615609757132981e-15, (11, 2) = -0.4728545352544275e-15, (11, 3) = -0.156166619860568e-15, (11, 4) = -0.3823761050108772e-11, (11, 5) = 0.26413002214707055e-10, (11, 6) = 0.3816857970803955e-11, (11, 7) = -0.26417699199310174e-10, (12, 1) = -0.9735840442745704e-15, (12, 2) = -0.11311490059501075e-15, (12, 3) = 0.572944480118278e-15, (12, 4) = 0.3129458491457907e-11, (12, 5) = -0.19617730829243156e-10, (12, 6) = -0.3139391932084518e-11, (12, 7) = 0.1962403285371077e-10, (13, 1) = 0.6993857106406554e-15, (13, 2) = 0.12016052989819686e-17, (13, 3) = -0.4775413954782384e-15, (13, 4) = 0.9776527453193262e-12, (13, 5) = -0.7353590626796982e-11, (13, 6) = -0.9732058285152592e-12, (13, 7) = 0.736398744908474e-11, (14, 1) = 0.9705949422791696e-15, (14, 2) = 0.2619714889118116e-15, (14, 3) = 0.4658921768483912e-15, (14, 4) = -0.11356829779398873e-11, (14, 5) = 0.859151329377958e-11, (14, 6) = 0.1130843856923472e-11, (14, 7) = -0.8594377426858616e-11, (15, 1) = 0.9695925570448124e-15, (15, 2) = -0.4491456893922899e-15, (15, 3) = -0.18981746494315082e-15, (15, 4) = -0.7483655333935907e-12, (15, 5) = 0.5705414294901479e-11, (15, 6) = 0.7510327670338783e-12, (15, 7) = -0.5693712029142713e-11, (16, 1) = -0.4211182412997506e-15, (16, 2) = 0.53675854807216605e-16, (16, 3) = 0.5636026768507778e-15, (16, 4) = -0.6995278907132813e-13, (16, 5) = 0.3956752285049062e-12, (16, 6) = 0.6435798249145133e-13, (16, 7) = -0.38962089425686327e-12, (17, 1) = -0.987797392137791e-15, (17, 2) = 0.1352320629979158e-15, (17, 3) = 0.736314984287521e-15, (17, 4) = 0.1600926011258469e-12, (17, 5) = -0.13596437821203659e-11, (17, 6) = -0.15449742937294287e-12, (17, 7) = 0.13567876892995205e-11, (18, 1) = 0.5480105428283639e-15, (18, 2) = 0.21991850644336722e-16, (18, 3) = -0.4957057157250095e-15, (18, 4) = 0.1115462835912193e-12, (18, 5) = -0.9972792947410631e-12, (18, 6) = -0.10002778723610967e-12, (18, 7) = 0.9946751007556866e-12, (19, 1) = 0.2051541063188908e-16, (19, 2) = 0.11294200854699044e-15, (19, 3) = 0.4045001749132188e-17, (19, 4) = 0.347646924947712e-13, (19, 5) = -0.29866875347509785e-12, (19, 6) = -0.26198740536541597e-13, (19, 7) = 0.2988957988372988e-12, (20, 1) = 0.12058191740209038e-15, (20, 2) = -0.38466585855218896e-16, (20, 3) = -0.2985194837894419e-15, (20, 4) = -0.434538460993134e-13, (20, 5) = 0.9256984699444391e-13, (20, 6) = 0.2636474583281246e-13, (20, 7) = -0.8881568135547719e-13, (21, 1) = -0.7236169266002019e-15, (21, 2) = 0.7490504568764222e-16, (21, 3) = 0.9003458239450704e-16, (21, 4) = -0.42293184595676174e-13, (21, 5) = 0.18141871038330725e-12, (21, 6) = 0.4753721029727625e-13, (21, 7) = -0.19446143748575735e-12, (22, 1) = 0.612944216527512e-15, (22, 2) = -0.8601158829868814e-16, (22, 3) = 0.1814914634218192e-15, (22, 4) = -0.4048428142474769e-13, (22, 5) = 0.12771648733499388e-12, (22, 6) = 0.33426089574883874e-13, (22, 7) = -0.1233125817553719e-12, (23, 1) = -0.6422347525704803e-15, (23, 2) = 0.54752741495183746e-16, (23, 3) = 0.1847519105913956e-15, (23, 4) = -0.23456109429317364e-13, (23, 5) = 0.599407945454169e-13, (23, 6) = 0.29480604199694505e-14, (23, 7) = -0.6198272077461873e-13, (24, 1) = 0.15289679052642838e-14, (24, 2) = 0.5952627674657526e-16, (24, 3) = -0.1466886454489566e-15, (24, 4) = -0.2467181232017062e-13, (24, 5) = 0.14429804392676375e-13, (24, 6) = 0.7489984412933763e-15, (24, 7) = -0.14288280747381178e-13, (25, 1) = 0.12086184087292001e-16, (25, 2) = 0.19839007361986904e-17, (25, 3) = -0.3622625788175379e-15, (25, 4) = -0.20921629902780556e-13, (25, 5) = -0.34397358966369926e-14, (25, 6) = -0.1491582276475305e-13, (25, 7) = -0.40828865281975146e-14, (26, 1) = 0.6835852097228387e-15, (26, 2) = 0.7159168873157178e-16, (26, 3) = -0.21106289557400055e-15, (26, 4) = -0.1906485271407097e-13, (26, 5) = -0.56163515551572984e-14, (26, 6) = -0.6276791431567554e-14, (26, 7) = -0.37098584055596615e-14, (27, 1) = 0.5350983365404941e-15, (27, 2) = 0.4692687628061364e-16, (27, 3) = -0.6738395569586877e-16, (27, 4) = -0.22590118680600477e-13, (27, 5) = -0.24220560752848112e-14, (27, 6) = 0.15201202620393297e-13, (27, 7) = -0.9850337523185942e-14, (28, 1) = 0.6386709760479533e-15, (28, 2) = 0.11665157737157613e-15, (28, 3) = -0.22481386616172177e-16, (28, 4) = -0.1493680309548479e-13, (28, 5) = 0.9588451217832344e-15, (28, 6) = 0.11332963376183775e-13, (28, 7) = 0.33988052418990065e-14, (29, 1) = -0.11858234993649612e-15, (29, 2) = 0.8345914577709899e-16, (29, 3) = -0.9440098605784009e-16, (29, 4) = -0.16354565186355035e-13, (29, 5) = 0.17313551359746578e-14, (29, 6) = -0.16694494925238307e-14, (29, 7) = 0.16415758210644425e-14, (30, 1) = 0.171441393554216e-15, (30, 2) = 0.7837222709221512e-16, (30, 3) = 0.2164837291089901e-16, (30, 4) = -0.2264321159369269e-13, (30, 5) = 0.11761895256182855e-14, (30, 6) = 0.5634349748706015e-14, (30, 7) = -0.3534149139630331e-14, (31, 1) = -0.8215363511663612e-15, (31, 2) = -0.8879314414573698e-16, (31, 3) = -0.3773725259825596e-16, (31, 4) = -0.3993302566613004e-14, (31, 5) = 0.12125877008205722e-14, (31, 6) = -0.2368948353793572e-14, (31, 7) = 0.5880139288555083e-14, (32, 1) = -0.14884070498466468e-14, (32, 2) = -0.9124595703922987e-16, (32, 3) = 0.3020809891896989e-16, (32, 4) = -0.13143876810222878e-13, (32, 5) = -0.4132395341286655e-15, (32, 6) = 0.1473633260802614e-13, (32, 7) = 0.2090468006401246e-14, (33, 1) = -0.17543347482846228e-15, (33, 2) = -0.2154896905096918e-15, (33, 3) = 0.3277997465846779e-17, (33, 4) = -0.13440643398875835e-13, (33, 5) = 0.7844345391382258e-15, (33, 6) = 0.43338191122353715e-14, (33, 7) = 0.4071948599435509e-14, (34, 1) = -0.10484289147259818e-14, (34, 2) = -0.17440369525656003e-16, (34, 3) = 0.14957519822825314e-17, (34, 4) = -0.11929388857164827e-13, (34, 5) = -0.884538010773651e-15, (34, 6) = 0.28384862076179502e-13, (34, 7) = 0.2909564106443845e-14, (35, 1) = -0.4743881179986118e-15, (35, 2) = -0.5749116292898412e-16, (35, 3) = -0.1754538341109995e-16, (35, 4) = -0.7260612619248704e-14, (35, 5) = -0.7983117465531899e-15, (35, 6) = -0.4565597313269818e-13, (35, 7) = -0.555817839560379e-15, (36, 1) = -0.1174552536015071e-14, (36, 2) = -0.9414186553278279e-16, (36, 3) = -0.3470056659136005e-16, (36, 4) = -0.16422018332342632e-13, (36, 5) = 0.18730988828982756e-14, (36, 6) = 0.24900215145384464e-13, (36, 7) = -0.3372457927277123e-14, (37, 1) = 0.11396594122078493e-14, (37, 2) = -0.9283946062547746e-16, (37, 3) = -0.4337891730736492e-16, (37, 4) = -0.16280450036247132e-13, (37, 5) = 0.34196034552693214e-14, (37, 6) = -0.8871495034418015e-14, (37, 7) = -0.3681530126204683e-14, (38, 1) = 0.9377121015665997e-15, (38, 2) = -0.15438058660056833e-15, (38, 3) = -0.664481589180503e-16, (38, 4) = -0.18207235377002977e-13, (38, 5) = 0.4537458614725343e-14, (38, 6) = 0.4479713894084907e-13, (38, 7) = -0.55983310867949246e-14, (39, 1) = 0.4912757437012582e-15, (39, 2) = -0.23729508346795716e-15, (39, 3) = -0.5127772136344247e-16, (39, 4) = -0.13283495289022541e-13, (39, 5) = 0.3902461771456388e-14, (39, 6) = 0.5637156253928188e-13, (39, 7) = -0.31979438060063607e-14, (40, 1) = 0.15079341456144565e-14, (40, 2) = 0.3686688323835932e-16, (40, 3) = -0.3437376752514623e-16, (40, 4) = -0.11767820879047535e-13, (40, 5) = 0.27339597378256296e-14, (40, 6) = -0.24655330087554253e-13, (40, 7) = -0.20908907585011148e-14, (41, 1) = 0.10863963128517745e-14, (41, 2) = -0.34928118246384267e-15, (41, 3) = -0.16841968462072937e-16, (41, 4) = -0.15357846249441516e-13, (41, 5) = -0.4362141498117359e-15, (41, 6) = 0.5448175863419239e-15, (41, 7) = -0.27993904926207662e-15, (42, 1) = -0.7491089899334664e-15, (42, 2) = -0.2389607311190943e-15, (42, 3) = 0.13687475637080167e-17, (42, 4) = -0.12033674311622374e-13, (42, 5) = -0.18861520099703315e-14, (42, 6) = -0.22102343527071885e-13, (42, 7) = 0.8463440413748968e-15, (43, 1) = 0.680271804198852e-15, (43, 2) = 0.105084827985502e-15, (43, 3) = 0.11914546778899865e-16, (43, 4) = -0.16605296131676905e-13, (43, 5) = -0.2094486937562489e-14, (43, 6) = -0.20032445900402725e-13, (43, 7) = 0.9981097724002823e-15, (44, 1) = 0.2561087537690394e-16, (44, 2) = 0.9104640411385184e-16, (44, 3) = 0.14321639187197174e-16, (44, 4) = -0.11097226283400303e-13, (44, 5) = -0.16169910985500647e-14, (44, 6) = 0.15034571829693287e-13, (44, 7) = -0.23371675220228955e-14, (45, 1) = 0.6862745553371683e-15, (45, 2) = -0.27886241130260373e-15, (45, 3) = 0.12741987354033401e-16, (45, 4) = -0.10468020370708285e-13, (45, 5) = -0.10645930056908387e-14, (45, 6) = -0.6650039209635444e-14, (45, 7) = 0.4132909992088207e-14, (46, 1) = -0.31344953050396397e-15, (46, 2) = -0.12019861400322722e-16, (46, 3) = 0.8951837113012151e-17, (46, 4) = -0.13640398673975572e-13, (46, 5) = -0.34518151355630866e-15, (46, 6) = -0.3521769796746239e-13, (46, 7) = 0.4757299655665785e-15, (47, 1) = -0.14978105381227495e-15, (47, 2) = -0.22532382147156626e-15, (47, 3) = 0.4983148361405653e-17, (47, 4) = -0.13988086974771912e-13, (47, 5) = -0.29420454951669475e-15, (47, 6) = 0.26487847805288282e-13, (47, 7) = -0.322905235030517e-15, (48, 1) = -0.29140372363198585e-15, (48, 2) = -0.16509515048737079e-15, (48, 3) = 0.18273691856145873e-17, (48, 4) = -0.7855787366784889e-14, (48, 5) = 0.10037992859786664e-14, (48, 6) = -0.13209515881504127e-13, (48, 7) = 0.17511195102865865e-14, (49, 1) = 0.9856271009687498e-16, (49, 2) = 0.7020545842080767e-16, (49, 3) = 0.14270528696284501e-18, (49, 4) = -0.13498233592649205e-13, (49, 5) = 0.12272534379896715e-14, (49, 6) = 0.17318737208180135e-14, (49, 7) = 0.7723675599972985e-14, (50, 1) = 0.9833432058197473e-15, (50, 2) = -0.2369335327431373e-15, (50, 3) = -0.7567388193426514e-18, (50, 4) = -0.11625921132393966e-13, (50, 5) = 0.13803409954644411e-14, (50, 6) = -0.11446236765334137e-13, (50, 7) = 0.26526350073863383e-14, (51, 1) = 0.5617416706181065e-15, (51, 2) = -0.4218222526508719e-15, (51, 3) = -0.11078221738943282e-17, (51, 4) = -0.9068896279418697e-14, (51, 5) = 0.16128282873163793e-14, (51, 6) = 0.2223679989069265e-13, (51, 7) = 0.626595209268804e-14, (52, 1) = -0.33490855756963006e-15, (52, 2) = -0.25443391856192324e-15, (52, 3) = -0.1144610906728241e-17, (52, 4) = -0.7279214270891835e-14, (52, 5) = 0.22819781770326752e-14, (52, 6) = 0.8344957490948999e-14, (52, 7) = -0.8382485001230783e-14, (53, 1) = 0.2487153554442684e-15, (53, 2) = -0.21293213376964732e-15, (53, 3) = -0.10189606213613277e-17, (53, 4) = -0.10323590373251864e-13, (53, 5) = 0.16711336104935634e-14, (53, 6) = 0.1700882708090868e-13, (53, 7) = -0.31761118210779397e-14, (54, 1) = 0.8135088044875406e-16, (54, 2) = 0.4178296263775635e-15, (54, 3) = -0.8398477733425834e-18, (54, 4) = -0.8578038014140092e-14, (54, 5) = 0.28458553066876643e-14, (54, 6) = -0.10008530986672623e-13, (54, 7) = -0.40245277618767896e-14, (55, 1) = -0.1159410563994957e-15, (55, 2) = -0.16403482084224275e-15, (55, 3) = -0.6296026412435251e-18, (55, 4) = -0.8665407265764291e-14, (55, 5) = 0.29543372771684898e-14, (55, 6) = 0.3559086333058728e-14, (55, 7) = -0.8595474785773308e-14, (56, 1) = -0.27247256436656254e-15, (56, 2) = -0.3493090176639956e-15, (56, 3) = -0.4496227051859683e-18, (56, 4) = -0.10618268747552274e-13, (56, 5) = 0.27234228753677925e-14, (56, 6) = 0.1221366473038268e-13, (56, 7) = -0.5304268341464851e-14, (57, 1) = 0.29909195107752526e-15, (57, 2) = -0.18524042296448685e-15, (57, 3) = -0.31185135444062216e-18, (57, 4) = -0.6435427389143579e-14, (57, 5) = 0.2396805761475413e-14, (57, 6) = -0.5100470128727699e-14, (57, 7) = 0.8165880176704672e-14, (58, 1) = 0.1504854797431194e-15, (58, 2) = 0.32830546870488394e-15, (58, 3) = -0.19638071141996554e-18, (58, 4) = -0.8069526748815397e-14, (58, 5) = 0.34075549514178215e-14, (58, 6) = 0.193994607658318e-14, (58, 7) = -0.14054076373514986e-13, (59, 1) = -0.8649850265936484e-16, (59, 2) = 0.1658790601892031e-15, (59, 3) = -0.12010111575595193e-18, (59, 4) = -0.7487090798726898e-14, (59, 5) = 0.32099877801629084e-14, (59, 6) = 0.11056309949757716e-13, (59, 7) = -0.19191590122252196e-14, (60, 1) = -0.17619086574072528e-15, (60, 2) = -0.9996036226992396e-16, (60, 3) = -0.6481918187265198e-19, (60, 4) = -0.5235506231335589e-14, (60, 5) = 0.4646277094210006e-14, (60, 6) = -0.62279302206787635e-15, (60, 7) = 0.68433730131112525e-14, (61, 1) = 0.1271898658122388e-15, (61, 2) = 0.34773037511140083e-15, (61, 3) = -0.26673901328089205e-19, (61, 4) = -0.9879548245473113e-14, (61, 5) = 0.5175399903587286e-14, (61, 6) = 0.11025529389594434e-13, (61, 7) = -0.10462758667135266e-13, (62, 1) = -0.50408180104168026e-16, (62, 2) = 0.1940407872481365e-15, (62, 3) = 0.11789886008640083e-22, (62, 4) = -0.53397256125135906e-14, (62, 5) = 0.3938989678411372e-14, (62, 6) = 0.9180019678055426e-14, (62, 7) = -0.12885252661787564e-13, (63, 1) = 0.21353164528158882e-15, (63, 2) = -0.16379675709012386e-15, (63, 3) = 0.12744325847619737e-19, (63, 4) = -0.4069432190171957e-14, (63, 5) = 0.5193321092500608e-14, (63, 6) = 0.6481356449364512e-14, (63, 7) = -0.1448840050123822e-13, (64, 1) = 0.6679785556111638e-16, (64, 2) = -0.5592410857669335e-16, (64, 3) = 0.22496656046256592e-19, (64, 4) = -0.42096472389684135e-14, (64, 5) = 0.6854577497542866e-14, (64, 6) = 0.2383116789478909e-14, (64, 7) = 0.29325306320331705e-14, (65, 1) = 0.24366408605403893e-16, (65, 2) = -0.1862704841413038e-15, (65, 3) = 0.248786340796678e-19, (65, 4) = -0.436757412314998e-14, (65, 5) = 0.11048773576552292e-13, (65, 6) = 0.4690594173981698e-14, (65, 7) = -0.6862647598267446e-14, (66, 1) = -0.30793139033254785e-17, (66, 2) = 0.8327453581489639e-16, (66, 3) = 0.25606413535867968e-19, (66, 4) = -0.4706235669008947e-14, (66, 5) = 0.9211988751244868e-14, (66, 6) = 0.4181706005465426e-14, (66, 7) = -0.6457844193638751e-14, (67, 1) = 0.27496551558527104e-16, (67, 2) = 0.6527029902579181e-16, (67, 3) = 0.236202191088386e-19, (67, 4) = -0.22205963412707087e-14, (67, 5) = 0.12934671651686996e-13, (67, 6) = 0.710764137271856e-15, (67, 7) = -0.19728597212423128e-13, (68, 1) = -0.18716928432831737e-16, (68, 2) = 0.1207343583753263e-15, (68, 3) = 0.2107478073615592e-19, (68, 4) = -0.7478843263438665e-15, (68, 5) = 0.10046806121547797e-13, (68, 6) = 0.15990052869078605e-15, (68, 7) = -0.15759291735764702e-13, (69, 1) = .0, (69, 2) = -0.12787666010188075e-15, (69, 3) = 0.18289766044212035e-19, (69, 4) = .0, (69, 5) = 0.23868102608592365e-14, (69, 6) = .0, (69, 7) = -0.9345256521634723e-14}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[69] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(2.6417699199310174e-11) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [7, 69, [f(x), diff(f(x), x), diff(diff(f(x), x), x), g(x), diff(g(x), x), h(x), diff(h(x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[69] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[69] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(7, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(69, 7, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(7, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(69, 7, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), g(x), diff(g(x), x), h(x), diff(h(x), x)]'[i] = yout[i], i = 1 .. 7)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[69] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(2.6417699199310174e-11) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [7, 69, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[69] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[69] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(69, 7, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(7, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0., (6) = 0., (7) = 0.}); `dsolve/numeric/hermite`(69, 7, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 7)] end proc, (2) = Array(0..0, {}), (3) = [x, f(x), diff(f(x), x), diff(diff(f(x), x), x), g(x), diff(g(x), x), h(x), diff(h(x), x)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), g(x), diff(g(x), x), h(x), diff(h(x), x)]'[i] = res[i+1], i = 1 .. 7)] catch: error  end try end proc

(4)

# Plots

display(
  odeplot(SOL, [x, f(x)], x=0..L, color=red, legend=typeset('f(x)'))
  , odeplot(SOL, [x, g(x)], x=0..L, color=green, legend=typeset('g(x)'))
  , odeplot(SOL, [x, h(x)], x=0..L, color=blue, legend=typeset('h(x)'))
)

 

# REMARK:
#
# This naive method doesn't always work


L := 100:
`ICs+BCs` :=  f(0)=0, D(f)(0)=1, g(0)=1, h(0)=1, f(L)=0, g(L)=0, h(L)=0;

SYS := {eq1, eq2, eq3, `ICs+BCs`}:
SOL := dsolve(eval(SYS, params=~data), numeric, maxmesh=10^4)

f(0) = 0, (D(f))(0) = 1, g(0) = 1, h(0) = 1, f(100) = 0, g(100) = 0, h(100) = 0

 

Error, (in dsolve/numeric/bvp) Newton iteration is not converging

 

 

A BETTER APPROACH: RE-PARAMETERIZE THE PROBLEM

 

# Method 2
#
# Re-parameterize the problem by using a 1-to-1 map from (0, +oo) into (0, 1).
#
# Here I use the change phi := x -> 1/(1-x) in such a way that +oo is mapped to 1/

edo := eval(eqs, {f(x)=f(phi(x)), g(x)=g(phi(x)), h(x)=h(phi(x))}):

convert(select(has, indets(edo, function), D), list):
Changes := % =~ subsop~(-1=z, %):
eval(eval(edo, Changes), {f(phi(x))=f(z), g(phi(x))=g(z), h(phi(x))=h(z)}):

EDO := (numer@lhs@simplify)~(convert(eval(eval(%, phi=(x->1/(1-x))), x=z), diff));

{g(z)*sin(alpha)*f(z)*z^4-4*g(z)*sin(alpha)*f(z)*z^3+6*g(z)*sin(alpha)*f(z)*z^2+(diff(g(z), z))*cos(alpha)*z^3-4*g(z)*sin(alpha)*f(z)*z-2*(diff(g(z), z))*cos(alpha)*z^2+sin(alpha)*f(z)*g(z)+cos(alpha)*z*(diff(g(z), z))+(diff(g(z), z))*z^2+(diff(g(z), z))*(diff(h(z), z))-4*(diff(g(z), z))*z+diff(diff(g(z), z), z)+3*(diff(g(z), z)), cos(alpha)*z^5+sin(alpha)*f(z)*z^4-4*cos(alpha)*z^4-4*sin(alpha)*f(z)*z^3+6*cos(alpha)*z^3+6*sin(alpha)*f(z)*z^2-4*cos(alpha)*z^2-4*sin(alpha)*f(z)*z-4*(diff(g(z), z))*z-4*(diff(h(z), z))*z+cos(alpha)*z+sin(alpha)*f(z)+2*(diff(diff(g(z), z), z))+4*(diff(g(z), z))+2*(diff(diff(h(z), z), z))+4*(diff(h(z), z)), -2*(diff(f(z), z))*cos(alpha)*z^4-2*(diff(f(z), z))*sin(alpha)*f(z)*z^3+(diff(diff(f(z), z), z))*cos(alpha)*z^3+(diff(diff(f(z), z), z))*sin(alpha)*f(z)*z^2+6*(diff(f(z), z))*cos(alpha)*z^3+6*(diff(f(z), z))*sin(alpha)*f(z)*z^2-2*(diff(diff(f(z), z), z))*cos(alpha)*z^2-2*(diff(diff(f(z), z), z))*sin(alpha)*f(z)*z-6*(diff(f(z), z))*cos(alpha)*z^2-6*(diff(f(z), z))*sin(alpha)*f(z)*z+(diff(diff(f(z), z), z))*cos(alpha)*z+(diff(diff(f(z), z), z))*sin(alpha)*f(z)+2*(diff(f(z), z))*cos(alpha)*z+2*(diff(f(z), z))*sin(alpha)*f(z)+12*(diff(f(z), z))*z^2-12*(diff(diff(f(z), z), z))*z-24*(diff(f(z), z))*z+2*(diff(diff(diff(f(z), z), z), z))+12*(diff(diff(f(z), z), z))+12*(diff(f(z), z))}

(5)

`ICs+BCs` :=  f(0)=0, D(f)(0)=1, g(0)=1, h(0)=1, f(1)=0, g(1)=0, h(1)=0;

SYS := {EDO[], `ICs+BCs`}:
SOL := dsolve(eval(SYS, params=~data), numeric, maxmesh=10^4)

f(0) = 0, (D(f))(0) = 1, g(0) = 1, h(0) = 1, (D(f))(1) = 0, (D(g))(1) = 0, (D(h))(1) = 0

 

f(0) = 0, (D(f))(0) = 1, g(0) = 1, h(0) = 1, f(1) = 0, g(1) = 0, h(1) = 0

 

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(14, {(1) = .0, (2) = 0.3867790186368738e-1, (3) = 0.7921400524698205e-1, (4) = .12190501921111957, (5) = .1675218836711356, (6) = .2194193635372026, (7) = .2819056705585553, (8) = .36151815970504464, (9) = .4737072735309197, (10) = .6374381555902743, (11) = .8114921352264324, (12) = .9041120924428426, (13) = .9583535448150223, (14) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(14, 7, {(1, 1) = .0, (1, 2) = 1.0, (1, 3) = -5.761848968646005, (1, 4) = 1.0, (1, 5) = -1.337031493173105, (1, 6) = 1.0, (1, 7) = -2.36496207850223, (2, 1) = 0.3462963925320743e-1, (2, 2) = .7970843110361767, (2, 3) = -4.765699307207549, (2, 4) = .9489455727555449, (2, 5) = -1.3023902740882791, (2, 6) = .9131938428115864, (2, 7) = -2.1295928907673263, (3, 1) = 0.6326837508749135e-1, (3, 2) = .6216048054057983, (3, 3) = -3.922746681703943, (3, 4) = .8969369551067902, (3, 5) = -1.2633063235581794, (3, 6) = .8312732294940347, (3, 7) = -1.9176983070509037, (4, 1) = 0.8645910211026682e-1, (4, 2) = .469899907557101, (4, 3) = -3.2110319102778644, (4, 4) = .8439116066992878, (4, 5) = -1.2207244374468957, (4, 6) = .753588351450841, (4, 7) = -1.7267149826330557, (5, 1) = .10477549194590889, (5, 2) = .33774119472791586, (5, 3) = -2.607008192089067, (5, 4) = .7892663008969169, (5, 5) = -1.175237463317485, (5, 6) = .6788947376396295, (5, 7) = -1.5528177610710407, (6, 1) = .11904899780493274, (6, 2) = .2169509270529728, (6, 3) = -2.0714414681725044, (6, 4) = .7295839052556613, (6, 5) = -1.1252041968075819, (6, 6) = .6027674137014565, (6, 7) = -1.3858662566991613, (7, 1) = .12890140956749632, (7, 2) = .10343166682049838, (7, 3) = -1.5870132660067657, (7, 4) = .6610495017098316, (7, 5) = -1.0693066830411293, (7, 6) = .521541142985605, (7, 7) = -1.2195400017914464, (8, 1) = .13261253507165302, (8, 2) = -0.4427616442988886e-2, (8, 3) = -1.1506792763262106, (8, 4) = .5784650774816225, (8, 5) = -1.0072007418365487, (8, 6) = .43148545786120995, (8, 7) = -1.0495995439341332, (9, 1) = .12580323122148082, (9, 2) = -.10970743762781049, (9, 3) = -.7607839575368128, (9, 4) = .469488948554787, (9, 5) = -.9394941949568495, (9, 6) = .32447631704681884, (9, 7) = -.8673887586250898, (10, 1) = 0.9924220043231033e-1, (10, 2) = -.20668614176469222, (10, 3) = -.46276289781044955, (10, 4) = .3209551479823555, (10, 5) = -.8831120502664324, (10, 6) = .1987896633674789, (10, 7) = -.6802682509267182, (11, 1) = 0.5710071898080872e-1, (11, 2) = -.2737116590716575, (11, 3) = -.3290308610212812, (11, 4) = .16867366804717226, (11, 5) = -.8753381202326455, (11, 6) = 0.9287644388323756e-1, (11, 7) = -.545309895620271, (12, 1) = 0.3038695568927371e-1, (12, 2) = -.3026989809224393, (12, 3) = -.3006639789969527, (12, 4) = 0.8694074852034647e-1, (12, 5) = -.8919295197045717, (12, 6) = 0.44931641966561915e-1, (12, 7) = -.4917973442026045, (13, 1) = 0.13530102428541772e-1, (13, 2) = -.31878506729557726, (13, 3) = -.293492335971878, (13, 4) = 0.3814181324271095e-1, (13, 5) = -.9081985353427808, (13, 6) = 0.18991126536757223e-1, (13, 7) = -.4652444582811626, (14, 1) = .0, (14, 2) = -.3309641620970596, (14, 3) = -.2919208315203016, (14, 4) = .0, (14, 5) = -.9239731614386971, (14, 6) = .0, (14, 7) = -.44708976653471394}, datatype = float[8], order = C_order); YP := Matrix(14, 7, {(1, 1) = 1.0, (1, 2) = -5.761848968646005, (1, 3) = 28.571093811876025, (1, 4) = -1.337031493173105, (1, 5) = .8490657004017081, (1, 6) = -2.36496207850223, (1, 7) = 6.554921442948961, (2, 1) = .7970843110361767, (2, 2) = -4.765699307207549, (2, 3) = 23.143280709518116, (2, 4) = -1.3023902740882791, (2, 5) = .9355996904458164, (2, 6) = -2.1295928907673263, (2, 7) = 5.641515892289439, (3, 1) = .6216048054057983, (3, 2) = -3.922746681703943, (3, 3) = 18.62300791661861, (3, 4) = -1.2633063235581794, (3, 5) = .9864356154450369, (3, 6) = -1.9176983070509037, (3, 7) = 4.837095223230097, (4, 1) = .469899907557101, (4, 2) = -3.2110319102778644, (4, 3) = 14.871145073630641, (4, 4) = -1.2207244374468957, (4, 5) = 1.002715038066246, (4, 6) = -1.7267149826330557, (4, 7) = 4.132342828102045, (5, 1) = .33774119472791586, (5, 2) = -2.607008192089067, (5, 3) = 11.743889510324548, (5, 4) = -1.175237463317485, (5, 5) = .9865506577147749, (5, 6) = -1.5528177610710407, (5, 7) = 3.5126344731119827, (6, 1) = .2169509270529728, (6, 2) = -2.0714414681725044, (6, 3) = 9.023889519177429, (6, 4) = -1.1252041968075819, (6, 5) = .936981722719509, (6, 6) = -1.3858662566991613, (6, 7) = 2.9426022070081155, (7, 1) = .10343166682049838, (7, 2) = -1.5870132660067657, (7, 3) = 6.614456551402488, (7, 4) = -1.0693066830411293, (7, 5) = .8479819019203072, (7, 6) = -1.2195400017914464, (7, 7) = 2.404562293326096, (8, 1) = -0.4427616442988886e-2, (8, 2) = -1.1506792763262106, (8, 3) = 4.4912958350942125, (8, 4) = -1.0072007418365487, (8, 5) = .7090694743244996, (8, 6) = -1.0495995439341332, (8, 7) = 1.8918870075283722, (9, 1) = -.10970743762781049, (9, 2) = -.7607839575368128, (9, 3) = 2.6286004910422314, (9, 4) = -.9394941949568495, (9, 5) = .4970064145266089, (9, 6) = -.8673887586250898, (9, 7) = 1.391013433521155, (10, 1) = -.20668614176469222, (10, 2) = -.46276289781044955, (10, 3) = 1.1869262345545342, (10, 4) = -.8831120502664324, (10, 5) = .1952163977719117, (10, 6) = -.6802682509267182, (10, 7) = .9347306060237894, (11, 1) = -.2737116590716575, (11, 2) = -.3290308610212812, (11, 3) = .43424396662998066, (11, 4) = -.8753381202326455, (11, 5) = -.10258104005182966, (11, 6) = -.545309895620271, (11, 7) = .6378805240815804, (12, 1) = -.3026989809224393, (12, 2) = -.3006639789969527, (12, 3) = .19052695806430253, (12, 4) = -.8919295197045717, (12, 5) = -.25539134969232746, (12, 6) = -.4917973442026045, (12, 7) = .520734967789831, (13, 1) = -.31878506729557726, (13, 2) = -.293492335971878, (13, 3) = 0.768018235889989e-1, (13, 4) = -.9081985353427808, (13, 5) = -.34449699165846415, (13, 6) = -.4652444582811626, (13, 7) = .45889425986623156, (14, 1) = -.3309641620970596, (14, 2) = -.2919208315203016, (14, 3) = -0.13766765505351941e-13, (14, 4) = -.9239731614386971, (14, 5) = -.41309894503196887, (14, 6) = -.44708976653471394, (14, 7) = .41309894503196865}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(14, {(1) = .0, (2) = 0.3867790186368738e-1, (3) = 0.7921400524698205e-1, (4) = .12190501921111957, (5) = .1675218836711356, (6) = .2194193635372026, (7) = .2819056705585553, (8) = .36151815970504464, (9) = .4737072735309197, (10) = .6374381555902743, (11) = .8114921352264324, (12) = .9041120924428426, (13) = .9583535448150223, (14) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(14, 7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = 0.20598739040450738e-6, (1, 4) = .0, (1, 5) = 0.21923986052295887e-7, (1, 6) = .0, (1, 7) = -0.36416628409076986e-7, (2, 1) = -0.2619181661824391e-9, (2, 2) = 0.9357612102628342e-8, (2, 3) = 0.15074386840527264e-6, (2, 4) = 0.8492866901865938e-9, (2, 5) = 0.1964365912012341e-7, (2, 6) = -0.14055296822321579e-8, (2, 7) = -0.3302631060007876e-7, (3, 1) = -0.13039753580884707e-9, (3, 2) = 0.1675304991495431e-7, (3, 3) = 0.10653797473776964e-6, (3, 4) = 0.1640711163858139e-8, (3, 5) = 0.17441260428100345e-7, (3, 6) = -0.27383842346694344e-8, (3, 7) = -0.29768773845633e-7, (4, 1) = 0.3481450255609154e-9, (4, 2) = 0.22491595264437804e-7, (4, 3) = 0.715812826185929e-7, (4, 4) = 0.23747992554424157e-8, (4, 5) = 0.15337941622927633e-7, (4, 6) = -0.4000502151994394e-8, (4, 7) = -0.26669807781286407e-7, (5, 1) = 0.11312762516720328e-8, (5, 2) = 0.26934999354014723e-7, (5, 3) = 0.43800421015563873e-7, (5, 4) = 0.30581504591337564e-8, (5, 5) = 0.13333165952851503e-7, (5, 6) = -0.520466199483298e-8, (5, 7) = -0.23722045273242863e-7, (6, 1) = 0.21598129125211313e-8, (6, 2) = 0.3093513827894949e-7, (6, 3) = 0.1879179571970469e-7, (6, 4) = 0.37163044694821774e-8, (6, 5) = 0.11349716994435048e-7, (6, 6) = -0.6404067353135797e-8, (6, 7) = -0.20798719930938432e-7, (7, 1) = 0.3307512119467471e-8, (7, 2) = 0.35610864799538496e-7, (7, 3) = -0.7974291257133687e-8, (7, 4) = 0.4327874412410117e-8, (7, 5) = 0.939717307144475e-8, (7, 6) = -0.7589736161622098e-8, (7, 7) = -0.17882365081628894e-7, (8, 1) = 0.38589114959125225e-8, (8, 2) = 0.44462865946981345e-7, (8, 3) = -0.4738889340171041e-7, (8, 4) = 0.4717687182962939e-8, (8, 5) = 0.7724717863007847e-8, (8, 6) = -0.852917136661407e-8, (8, 7) = -0.1526132366423764e-7, (9, 1) = -0.28884318368283274e-8, (9, 2) = 0.8293977419753843e-7, (9, 3) = -0.16615143597804478e-6, (9, 4) = 0.35662271991628005e-8, (9, 5) = 0.8111828099503108e-8, (9, 6) = -0.7139294024166424e-8, (9, 7) = -0.1526015865346634e-7, (10, 1) = -0.4782880287140578e-7, (10, 2) = 0.23412011256377628e-6, (10, 3) = -0.4495933404811711e-6, (10, 4) = -0.34189824447296726e-8, (10, 5) = 0.12992963820910588e-7, (10, 6) = 0.5375525946571291e-8, (10, 7) = -0.21338423790326948e-7, (11, 1) = -0.2889828366501854e-7, (11, 2) = 0.17055043277306273e-6, (11, 3) = -0.20549706111432103e-6, (11, 4) = -0.4735225854261932e-9, (11, 5) = 0.18621929893883327e-8, (11, 6) = 0.15805934508596413e-8, (11, 7) = -0.572857873018638e-8, (12, 1) = -0.13201704208102298e-7, (12, 2) = 0.14603036546603085e-6, (12, 3) = -0.18008550489606026e-6, (12, 4) = -0.10687258886534304e-9, (12, 5) = 0.9268257193941267e-9, (12, 6) = 0.5175045431967232e-9, (12, 7) = -0.4408009660340614e-8, (13, 1) = -0.5490366201406191e-8, (13, 2) = 0.13542362394229005e-6, (13, 3) = -0.17550190507188007e-6, (13, 4) = -0.4289230724173514e-10, (13, 5) = 0.9537603573503932e-9, (13, 6) = 0.20954546346126312e-9, (13, 7) = -0.43864846559118675e-8, (14, 1) = .0, (14, 2) = 0.12773631346883834e-6, (14, 3) = -0.17450565210332377e-6, (14, 4) = .0, (14, 5) = 0.10524400143852329e-8, (14, 6) = .0, (14, 7) = -0.4476298082660726e-8}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[14] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(4.495933404811711e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [7, 14, [f(z), diff(f(z), z), diff(diff(f(z), z), z), g(z), diff(g(z), z), h(z), diff(h(z), z)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[14] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[14] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(7, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(14, 7, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(7, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(14, 7, X, Y, outpoint, yout, L, V) end if; [z = outpoint, seq('[f(z), diff(f(z), z), diff(diff(f(z), z), z), g(z), diff(g(z), z), h(z), diff(h(z), z)]'[i] = yout[i], i = 1 .. 7)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[14] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(4.495933404811711e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [7, 14, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[14] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[14] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(7, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(14, 7, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(7, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0., (6) = 0., (7) = 0.}); `dsolve/numeric/hermite`(14, 7, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 7)] end proc, (2) = Array(0..0, {}), (3) = [z, f(z), diff(f(z), z), diff(diff(f(z), z), z), g(z), diff(g(z), z), h(z), diff(h(z), z)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [z = res[1], seq('[f(z), diff(f(z), z), diff(diff(f(z), z), z), g(z), diff(g(z), z), h(z), diff(h(z), z)]'[i] = res[i+1], i = 1 .. 7)] catch: error  end try end proc

(6)

display(
  odeplot(SOL, [1/(1-z), f(z)], z=0..0.999, axis[1]=[mode=log], color=red, legend=typeset('f(x)'), labels=[x, ""])
  , odeplot(SOL, [1/(1-z), g(z)], z=0..0.999, axis[1]=[mode=log], color=green, legend=typeset('g(x)'))
  , odeplot(SOL, [1/(1-z), h(z)], z=0..0.999, axis[1]=[mode=log], color=blue, legend=typeset('h(x)'))
)

 

 


Download BCs_at_infinity.mw

Change the title for somthong more meaningful

plot(
  (subs(M = 1.3015, A)-subs(M = 1.2431, A))/subs(M = 1.2431, A)*100
  , x = -5 .. 0
  , axes = boxed
  , colour = red
  , tickmarks=[default, [seq(k=sprintf("%d%%", k), k in [seq](-60..0, 10))]]
  , title="(A-Aref)/Aref*100%"
  , gridlines=true
)

 

NULL

Download percentage.mw

Neither differential equation 1 nor differential equation 2 are ODEs bur simply DEs.

IMO, the correct approach to handle this situation is to proceed this way (here illustrated on case 2 for simplicity):

ode:=diff(y(x),x)^2-(1+2*x*y(x))*diff(y(x),x)+2*x*y(x) = 0;
S := [solve(ode, diff(y(x), x))];
                         [1, 2 y(x) x]
map(s -> dsolve(diff(y(x), x)=s), S)
              [                              / 2\]
              [y(x) = x + _C1, y(x) = _C1 exp\x /]

By chance the solutions you got are identical to those above because the two "branches" 1 and 2*x*y(x) are easy to obtain and have simple expressions.

Let's go now to case 1. Here the "branches" have extremely complex expressions and Maple, quite logically, fails in finding the general solution.
Details are given in this attached file:

Download DE.mw

As you will see in there are four "branches" along which y(x) is governed by an ode of the form

diff(yn(x), x) = Fn(yn(x), x), n=1..4

You can observe that each Fn(yn(x), x) is singular at x=0 (which indicates that initial conditiopns for the form y(0)=C are inappropriate, unless you use a local series expansion to fix this problem).

From a numerical point of view dsolve/numeric seems to fail too (more precisely I wans't capable to find any "good" initial condition).

The "simpler" case I treat at the end of the file, is a case I came across during my professional work.It is not that far from your case 1 (no sqrt in my "simpler" case).

One is to build it from a Beta random variable, the other (more complex) is to define a SubjectiveBeta distribution.

Here is an incomplete example (I didn't introduce the Skewness, Kurtosis, and other statistics into the definition of SubjectiveBeta)

restart:

with(Statistics):


WAY 1:
          Define B ~ Beta(alpha, beta)
          And next BSD := x__min + (x__max-x__min)*B

B := RandomVariable('Beta' (alpha, beta))

_R

(1)

BSD := x__min + (x__max-x__min)*B

x__min+(x__max-x__min)*_R

(2)

# PDF of BSD

f := PDF(BSD, x)

f := piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+alpha)*(1-(x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+beta)/Beta(alpha, beta), 0)/abs(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)

(3)

# PDF of BSD in the form given in your reference

eval(f) assuming x > x__min, x__max > x__min, x < x__max:

subs(x__max-x__min=h, %):
map(simplify, %) assuming alpha > 0, beta > 0, h > 0:
simplify(%):

f__BSD := subs(h=x__max-x__min, %);
 

(x__max-x__min)^(-alpha+1-beta)*(x-x__min)^(-1+alpha)*(x__max-x)^(-1+beta)/Beta(alpha, beta)

(4)

# A closed form of the CDF does exist contraryto what is said in your reference

F := CDF(BSD, x);
print():

# After accounting for assumptions:

eval(F)  assuming x > x__min, x__max > x__min, x < x__max
 

F := piecewise(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))` < 0, 1-piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^alpha*hypergeom([alpha, 1-beta], [1+alpha], (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))/(Beta(alpha, beta)*alpha), 1), `#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))` = 0, piecewise(`#msub(mi("x"),mi("min",fontstyle = "normal"))` <= x, 1, 0), piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^alpha*hypergeom([alpha, 1-beta], [1+alpha], (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))/(Beta(alpha, beta)*alpha), 1))

 

 

((x-x__min)/(x__max-x__min))^alpha*hypergeom([alpha, 1-beta], [1+alpha], (x-x__min)/(x__max-x__min))/(Beta(alpha, beta)*alpha)

(5)

# Direct way (I don't know how to simplify this expression)


'mean(BSD)' = simplify(simplify(Mean(BSD), GAMMA), GAMMA);

# alternative way (using the linearity of the "Expectation" operator

'mean(BSD)' = x__min + (x__max-x__min)*Mean(B);

mean(BSD) = (alpha*x__max+beta*x__min)*GAMMA(alpha)*GAMMA(beta)/(Beta(alpha, beta)*GAMMA(alpha+beta+1))

 

mean(BSD) = x__min+(x__max-x__min)*alpha/(alpha+beta)

(6)

# Direct way (I don't know how to simplify this expression)

'variance(BSD)' =  simplify(Variance(BSD), GAMMA):

# alternative way (using the linearity of the "Expectation" operator

'variance(BSD)' = (x__max-x__min)^2*Variance(B);

variance(BSD) = (x__max-x__min)^2*alpha*beta/((alpha+beta)^2*(alpha+beta+1))

(7)

# Direct way (fails)

'mode(BSD)' = simplify(Mode(BSD), GAMMA);

# alternative way (obvious result)

'mode(BSD)' = x__min + (x__max-x__min)*Mode(B);

Warning, solutions may have been lost

 

"?={{Typesetting:-mi("t",italic = "true",mathvariant = "italic")}}"

 

mode(BSD) = x__min+(x__max-x__min)*piecewise(alpha < 1 and beta < 1, {0, 1}, 1 < alpha and 1 < beta, (-1+alpha)/(alpha+beta-2), alpha < 1 and 1 <= beta, 0, 1 <= alpha and beta < 1, 1, FAIL)

(8)


WAY 2:
          Define a BSD distribution

B   := RandomVariable('Beta' (alpha, beta)):
BSD := x__min + (x__max-x__min)*B:

# f and F above are used to be copied-pasted into SubjectiveBeta

f   := PDF(BSD, x);
F   := CDF(BSD, x) assuming x__max > x__min

f := piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+alpha)*(1-(x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+beta)/Beta(alpha, beta), 0)/abs(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)

 

piecewise((x-x__min)/(x__max-x__min) < 0, 0, (x-x__min)/(x__max-x__min) < 1, ((x-x__min)/(x__max-x__min))^alpha*hypergeom([alpha, 1-beta], [1+alpha], (x-x__min)/(x__max-x__min))/(Beta(alpha, beta)*alpha), 1)

(9)

SubjectiveBeta := proc(A, B, P, Q)
   Distribution(
     PDF=unapply(
       eval(
         piecewise(
           (x-x__min)/(x__max-x__min) < 0, 0
           , (x-x__min)/(x__max-x__min) < 1
           , ((x-x__min)/(x__max-x__min))^(-1+alpha)*(1-(x-x__min)/(x__max-x__min))^(-1+beta)/Beta(alpha, beta)
           , 0
         )/(x__max-x__min)
         , [alpha=A, beta=B, x__min=P, x__max=Q]
       )
       , t
     ),
     CDF=unapply(
       eval(
         piecewise(
           (x-x__min)/(x__max-x__min) < 0, 0
           , (x-x__min)/(x__max-x__min) < 1,
               ((x-x__min)/(x__max-x__min))^alpha
               *
               hypergeom([alpha, 1-beta], [alpha+1], (x-x__min)/(x__max-x__min))/(Beta(alpha, beta)*alpha)
           , 1
         )
         , [alpha=A, beta=B, x__min=P, x__max=Q]
       )
       , t
      ),

      Mean = eval(
               x__min+(x__max-x__min)*alpha/(alpha+beta)
               , [alpha=A, beta=B, x__min=P, x__max=Q]
             ),

      Variance = eval(
                   (x__max-x__min)^2*alpha*beta/((alpha+beta)^2*(alpha+beta+1))
                   , [alpha=A, beta=B, x__min=P, x__max=Q]
                 ),

      Mode = eval(
               x__min+(x__max-x__min) * Mode('Beta'(A, B))
               , [alpha=A, beta=B, x__min=P, x__max=Q]
             ),

      Conditions = [A > 0, B > 0, Q > P],

      RandomSample = proc(N::nonnegint)
                       P +~ (Q-P) *~ Sample('Beta'(A, B), N)
                     end proc
   )
end proc:

BSD := (alpha, beta, xmin, xmax) -> RandomVariable(SubjectiveBeta(alpha, beta, xmin, xmax)):

SubjBeta := BSD(alpha, beta, x__min, x__max);

_R6

(10)

PDF(SubjBeta, x);

CDF(SubjBeta, x);

Mean(SubjBeta);
Variance(SubjBeta);
Mode(SubjBeta);

piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+alpha)*(1-(x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^(-1+beta)/Beta(alpha, beta), 0)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)

 

piecewise((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 0, 0, (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`) < 1, ((x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))^alpha*hypergeom([alpha, 1-beta], [1+alpha], (x-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)/(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`))/(Beta(alpha, beta)*alpha), 1)

 

`#msub(mi("x"),mi("min",fontstyle = "normal"))`+(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)*alpha/(alpha+beta)

 

(`#msub(mi("x"),mi("max",fontstyle = "normal"))`-`#msub(mi("x"),mi("min",fontstyle = "normal"))`)^2*alpha*beta/((alpha+beta)^2*(alpha+beta+1))

 

x__min+(x__max-x__min)*piecewise(alpha < 1 and beta < 1, {0, 1}, 1 < alpha and 1 < beta, (-1+alpha)/(alpha+beta-2), alpha < 1 and 1 <= beta, 0, 1 <= alpha and beta < 1, 1, FAIL)

(11)

SubjBetaNum := BSD(3, 1, -1, 4);

PDF(SubjBetaNum, x);

CDF(SubjBetaNum, x);

Mean(SubjBetaNum);

Variance(SubjBetaNum);

Mode(SubjBetaNum);

SubjBetaNum := _R8

 

(1/5)*piecewise((1/5)*x < -1/5, 0, (1/5)*x < 4/5, 3*((1/5)*x+1/5)^2, 0)

 

piecewise((1/5)*x < -1/5, 0, (1/5)*x < 4/5, ((1/5)*x+1/5)^3, 1)

 

11/4

 

15/16

 

4

(12)

Histogram( Sample(SubjBetaNum, 1000) )

 

 

Download BSD.mw

It would have been possible to build a module instead of a procedure to define SubjectiveBeta but I don't know what is the most robust approach (I let the specialists answer this and complete my answer).

Here is a corrected version, but as you provide more bcs than required it's up to you to clean it up.

restart

kernelopts(version);

`Maple 2015.2, APPLE UNIVERSAL OSX, Dec 20 2015, Build ID 1097895`

(1)

with(plots):

eq1 := diff(f(x), `$`(x, 3))+(1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[fl])*f(x)*(diff(f(x), `$`(x, 2)))+(1-phi)^2.5*M*`sin&alpha;`^2*(1-(diff(f(x), x)))+(1-phi)^2.5*(1-phi+phi*`&rho;&beta;`[s]/`&rho;&beta;`[fl])*Gr[x]*theta(x) = 0;

diff(diff(diff(f(x), x), x), x)+(1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[fl])*f(x)*(diff(diff(f(x), x), x))+(1-phi)^2.5*M*`sin&alpha;`^2*(1-(diff(f(x), x)))+(1-phi)^2.5*(1-phi+phi*`&rho;&beta;`[s]/`&rho;&beta;`[fl])*Gr[x]*theta(x) = 0

(2)

eq2 := K[nf]*(diff(theta(x), `$`(x, 2)))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*C[p][s]/(rho[fl]*C[p][f]))*f(x)*(diff(theta(x), x)) = 0;

K[nf]*(diff(diff(theta(x), x), x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*C[p][s]/(rho[fl]*C[p][f]))*f(x)*(diff(theta(x), x)) = 0

(3)

bcs := f(0) = 0, (D(f))(0) = epsilon, (D(f))(10) = 1;

f(0) = 0, (D(f))(0) = epsilon, (D(f))(10) = 1

 

theta(0) = 0, (D(theta))(0) = -Bi*(1-theta(0)), theta(10) = 0

(4)

a1 := [phi = .1, rho[s] = 5200, rho[fl] = 997.1, `&rho;&beta;`[s] = 6500, `&rho;&beta;`[fl] = 20939.1, epsilon = 0, C[p][s] = 670, C[p][f] = 4179, Bi = .1, M = 1, `sin&alpha;` = 0, Gr[x] = .1, Pr = 6.2, eta = 7];

[phi = .1, rho[s] = 5200, rho[fl] = 997.1, `&rho;&beta;`[s] = 6500, `&rho;&beta;`[fl] = 20939.1, epsilon = 0, C[p][s] = 670, C[p][f] = 4179, Bi = .1, M = 1, `sin&alpha;` = 0, Gr[x] = .1, Pr = 6.2, eta = 7]

 

[phi = .1, rho[s] = 5200, rho[fl] = 997.1, `&rho;&beta;`[s] = 6500, `&rho;&beta;`[fl] = 20939.1, epsilon = .2, C[p][s] = 670, C[p][f] = 4179, Bi = .1, M = 1, `sin&alpha;` = (1/2)*2^(1/2), Gr[x] = .1, Pr = 6.2, eta = 7]

 

[phi = .1, rho[s] = 5200, rho[fl] = 997.1, `&rho;&beta;`[s] = 6500, `&rho;&beta;`[fl] = 20939.1, epsilon = -.2, C[p][s] = 670, C[p][f] = 4179, Bi = .1, M = 1, `sin&alpha;` = 1, Gr[x] = .1, Pr = 6.2, eta = 7]

(5)

# What you want to "dsolve" is:

ToDsolve := [ subs(a1,eq1), bcs, subs(a1,eq2), bcs1 ]:

print~(ToDsolve):
 

diff(diff(diff(f(x), x), x), x)+.5461688485*f(x)*(diff(diff(f(x), x), x))+0.7154441464e-1*theta(x) = 0

 

f(0) = 0

 

(D(f))(0) = epsilon

 

(D(f))(10) = 1

 

K[nf]*(diff(diff(theta(x), x), x))/K[f]+3.049196273*f(x)*(diff(theta(x), x)) = 0

 

theta(0) = 0

 

(D(theta))(0) = -Bi*(1-theta(0))

 

theta(10) = 0

(6)

# ToDsolve contains some formal quantities:

FQ := convert(indets(ToDsolve, name) minus {f, theta, x}, list)

[Bi, epsilon, K[f], K[nf]]

(7)

# Give them numerical values and look to what happens

dsolve(eval(ToDsolve, FQ=~1), numeric)

Error, (in dsolve/numeric/bvp/convertsys) too many boundary conditions: expected 5, got 6

 

# The message means that you have 3 bcs for theta as theta verifies a 2nd order ode
# It's upt to you to fix that.
# I just show you that once fixed thete is no longer any error:
#
# Attempt 1

attempt := subsop(7=NULL, ToDsolve):
print~(attempt ):
dsolve(eval(attempt , FQ=~1), numeric)

diff(diff(diff(f(x), x), x), x)+.5461688485*f(x)*(diff(diff(f(x), x), x))+0.7154441464e-1*theta(x) = 0

 

f(0) = 0

 

(D(f))(0) = epsilon

 

(D(f))(10) = 1

 

K[nf]*(diff(diff(theta(x), x), x))/K[f]+3.049196273*f(x)*(diff(theta(x), x)) = 0

 

theta(0) = 0

 

theta(10) = 0

 

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(8, {(1) = .0, (2) = 1.4285714285714282, (3) = 2.8571428571428568, (4) = 4.285714285714286, (5) = 5.714285714285715, (6) = 7.142857142857144, (7) = 8.571428571428571, (8) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(8, 5, {(1, 1) = .0, (1, 2) = 1.0, (1, 3) = -0.8748723552754536e-32, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.4285714285714282, (2, 2) = 1.0, (2, 3) = -0.5617824718494409e-32, (2, 4) = .0, (2, 5) = .0, (3, 1) = 2.8571428571428568, (3, 2) = 1.0, (3, 3) = -0.11760570538661568e-32, (3, 4) = .0, (3, 5) = .0, (4, 1) = 4.285714285714286, (4, 2) = 1.0, (4, 3) = 0.5823110966250819e-34, (4, 4) = .0, (4, 5) = .0, (5, 1) = 5.714285714285715, (5, 2) = 1.0, (5, 3) = -0.5329316534916845e-35, (5, 4) = .0, (5, 5) = .0, (6, 1) = 7.142857142857144, (6, 2) = 1.0, (6, 3) = 0.16117092548942904e-35, (6, 4) = .0, (6, 5) = .0, (7, 1) = 8.571428571428571, (7, 2) = 1.0, (7, 3) = -0.6101608827896203e-36, (7, 4) = .0, (7, 5) = .0, (8, 1) = 10.0, (8, 2) = 1.0, (8, 3) = 0.208504395860558e-36, (8, 4) = .0, (8, 5) = .0}, datatype = float[8], order = C_order); YP := Matrix(8, 5, {(1, 1) = 1.0, (1, 2) = -0.8748723552754536e-32, (1, 3) = .0, (1, 4) = .0, (1, 5) = -.0, (2, 1) = 1.0, (2, 2) = -0.5617824718494409e-32, (2, 3) = 0.43832583679641825e-32, (2, 4) = .0, (2, 5) = -.0, (3, 1) = 1.0, (3, 2) = -0.11760570538661568e-32, (3, 3) = 0.1835216362515375e-32, (3, 4) = .0, (3, 5) = -.0, (4, 1) = 1.0, (4, 2) = 0.5823110966250819e-34, (4, 3) = -0.1363029347624971e-33, (4, 4) = .0, (4, 5) = -.0, (5, 1) = 1.0, (5, 2) = -0.5329316534916845e-35, (5, 3) = 0.16632609572385963e-34, (5, 4) = .0, (5, 5) = -.0, (6, 1) = 1.0, (6, 2) = 0.16117092548942904e-35, (6, 3) = -0.6287609913302913e-35, (6, 4) = .0, (6, 5) = -.0, (7, 1) = 1.0, (7, 2) = -0.6101608827896203e-36, (7, 3) = 0.28564360007395747e-35, (7, 4) = .0, (7, 5) = -.0, (8, 1) = 1.0, (8, 2) = 0.208504395860558e-36, (8, 3) = -0.11387860579434913e-35, (8, 4) = .0, (8, 5) = -.0}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(8, {(1) = .0, (2) = 1.4285714285714282, (3) = 2.8571428571428568, (4) = 4.285714285714286, (5) = 5.714285714285715, (6) = 7.142857142857144, (7) = 8.571428571428571, (8) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(8, 5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = 0.1720275018820703e-31, (1, 4) = .0, (1, 5) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = 0.10762107357861248e-31, (2, 4) = .0, (2, 5) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = 0.21900843254972705e-32, (3, 4) = .0, (3, 5) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = -0.508259396377478e-34, (4, 4) = .0, (4, 5) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = 0.7139700174278538e-35, (5, 4) = .0, (5, 5) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = -0.21488562233419584e-35, (6, 4) = .0, (6, 5) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = 0.8135504827948244e-36, (7, 4) = .0, (7, 5) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = -0.27800549133287224e-36, (8, 4) = .0, (8, 5) = .0}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[8] elif outpoint = "order" then return 2 elif outpoint = "error" then return HFloat(1.720275018820703e-32) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [5, 8, [f(x), diff(f(x), x), diff(diff(f(x), x), x), theta(x), diff(theta(x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[8] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[8] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(5, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(8, 5, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(5, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(8, 5, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), theta(x), diff(theta(x), x)]'[i] = yout[i], i = 1 .. 5)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[8] elif outpoint = "order" then return 2 elif outpoint = "error" then return HFloat(1.720275018820703e-32) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [5, 8, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[8] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[8] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(8, 5, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(5, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0.}); `dsolve/numeric/hermite`(8, 5, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 5)] end proc, (2) = Array(0..0, {}), (3) = [x, f(x), diff(f(x), x), diff(diff(f(x), x), x), theta(x), diff(theta(x), x)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), theta(x), diff(theta(x), x)]'[i] = res[i+1], i = 1 .. 5)] catch: error  end try end proc

(8)

# Attempt 2

attempt := subsop(8=NULL, ToDsolve):
print~(attempt ):
dsolve(eval(attempt , FQ=~1), numeric)

diff(diff(diff(f(x), x), x), x)+.5461688485*f(x)*(diff(diff(f(x), x), x))+0.7154441464e-1*theta(x) = 0

 

f(0) = 0

 

(D(f))(0) = epsilon

 

(D(f))(10) = 1

 

K[nf]*(diff(diff(theta(x), x), x))/K[f]+3.049196273*f(x)*(diff(theta(x), x)) = 0

 

theta(0) = 0

 

(D(theta))(0) = -Bi*(1-theta(0))