tomleslie

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12 years, 231 days

MaplePrimes Activity


These are answers submitted by tomleslie

So you wantt a solution for a system of odes which contains some totally arbitrary function h(t)?

Just for amusement in the attached, I have shown that Maple will provide solutions for a variety of possibilities of this  otherwise completely indeterminate function - I used h(t)=0, h(t)=cos(t) and h(t)=t^2. In all cases, a solution is found. But for h(t) entirely arbitrary - get real!

You might want to consider the attached where I have used some more-or-less  *random* guesses for h(t), as in   h(t)=0,  h(t)=cos(t), and h(t)=t^2. All of these generate "solutions". But for h(t) entirely arbitrary - no chance!

  restart:
  assume(lambda>0);
  interface(showassumed=0):
  odeSys:= diff(x(t),t)=y(t),
           diff( y(t), t) =z(t),
           diff(z(t),t)=-lambda*y(t)-h(t);
  dsolve(eval( [odeSys], h(t)=0));
  dsolve(eval( [odeSys], h(t)=cos(t)));
  dsolve(eval( [odeSys], h(t)=t^2));
  

diff(x(t), t) = y(t), diff(y(t), t) = z(t), diff(z(t), t) = -lambda*y(t)-h(t)

 

{x(t) = (_C1*lambda^(1/2)-cos(lambda^(1/2)*t)*_C2+sin(lambda^(1/2)*t)*_C3)/lambda^(1/2), y(t) = _C2*sin(lambda^(1/2)*t)+_C3*cos(lambda^(1/2)*t), z(t) = lambda^(1/2)*(cos(lambda^(1/2)*t)*_C2-sin(lambda^(1/2)*t)*_C3)}

 

{x(t) = (_C1*lambda^(3/2)-lambda*cos(lambda^(1/2)*t)*_C3+lambda*sin(lambda^(1/2)*t)*_C2-sin(t)*lambda^(1/2)-_C1*lambda^(1/2)+_C3*cos(lambda^(1/2)*t)-_C2*sin(lambda^(1/2)*t))/(lambda^(1/2)*(lambda-1)), y(t) = sin(lambda^(1/2)*t)*_C3+cos(lambda^(1/2)*t)*_C2-cos(t)/(lambda-1), z(t) = (lambda^(1/2)*cos(lambda^(1/2)*t)*_C3*(lambda-1)-lambda^(1/2)*sin(lambda^(1/2)*t)*_C2*(lambda-1)+sin(t))/(lambda-1)}

 

{x(t) = -_C3*cos(lambda^(1/2)*t)/lambda^(1/2)+_C2*sin(lambda^(1/2)*t)/lambda^(1/2)-(1/3)*t^3/lambda+_C1+2*t/lambda^2, y(t) = (sin(lambda^(1/2)*t)*_C3*lambda^2+cos(lambda^(1/2)*t)*_C2*lambda^2-lambda*t^2+2)/lambda^2, z(t) = (lambda^(3/2)*cos(lambda^(1/2)*t)*_C3-lambda^(3/2)*sin(lambda^(1/2)*t)*_C2-2*t)/lambda}

(1)

 

Download badProb.mw

 

but probably the simplest is to use the surfdata() command as shown in the attached. Many options are available with the surfdata() command, none of which I have bothered with

Two points to note

  1. You will have to change the file path in th ExcelTools:-Import() command to something appropriate for your machine
  2. It is not obvious from the data yu supply which direction you regard as 'x' and which is 'y', so I madee an arbitrary choice

   restart;
   with(plots):
   f := ExcelTools:-Import("C:/Users/TomLeslie/Desktop/PlotTest.xlsx"):
   L:= [ seq
         ( [ seq
             ( [ f[i+1,1], f[1,j+1], f[i+1, j+1] ],
                 i=1..9
             )
           ],
           j=1..9
         )
       ]:
   surfdata(L);

 

 

NULL

Download surfplot.mw

to get a numerical solution for your problem - although it appears that Maple cannot find an analytic solution.

See the attached.

After the point of achieving a solution, I have no idea what your subsequent code is trying to achieve, so I have ignored it!

  restart:
  with(plots):
  _local(I):
  Digits := 15:
  odeSys:= (1 - p)*(diff(S(t), t) + mu*S(t)) + p*(diff(S(t), t) + mu*S(t) + beta*S(t)*I(t) - rho*R(t) - varepsilon),
           (1 - p)*(diff(E(t), t) + (alpha1 + mu)*E(t)) + p*(diff(E(t), t) + (alpha1 + mu)*E(t) - beta*S(t)*I(t)),
           (1 - p)*(diff(I(t), t) + (alpha2 + delta + mu)*I(t)) + p*(diff(I(t), t) + (alpha2 + delta + mu)*I(t) - alpha1*E(t)),
           (1 - p)*(diff(R(t), t) + (mu + rho)*R(t)) + p*(diff(R(t), t) + (mu + rho)*R(t) - alpha2*I(t)):
  params:= [ mu=0.133*10^(-5),
             varepsilon=0.99879,
             delta=0.004554,
             beta=0.1009*10^(-6),
             alpha1=0.0008999,
             alpha2=0.1997,
             rho=0.00090021,
             p=1
           ]:
  ibvc := S(0) = 2304219, E(0) = 84929, I(0) = 299, R(0) = 71411:
#
# Won't find an analytic solution :-(
#
  solA:=dsolve(eval( [odeSys ,ibvc], params));
#
# Will find a numeric solution :-)
#
  solN:=dsolve(eval( [odeSys,ibvc], params), numeric);
  odeplot(solN, [t, S(t)], t=0..10, color=[red, green, blue, black]);
  odeplot(solN, [t, E(t)], t=0..10, color=[red, green, blue, black]);
  odeplot(solN, [t, I(t)], t=0..10, color=[red, green, blue, black]);
  odeplot(solN, [t, R(t)], t=0..10, color=[red, green, blue, black]);

I

 

Warning, The imaginary unit, I, has been renamed _I

 

 

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 := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 28, [( 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..65, {(1) = 4, (2) = 4, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (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, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .9832164330847996, (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..4, {(1) = 84929.0, (2) = 299.0, (3) = 71411.0, (4) = 2304219.0}, datatype = float[8], order = C_order)), ( 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..4, {(1) = .1, (2) = .1, (3) = .1, (4) = .1}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, 1..4, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0}, datatype = float[8], order = C_order), Array(1..4, 1..4, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, 1..4, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0}, datatype = float[8], order = C_order), Array(1..4, 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}, datatype = float[8], order = C_order), Array(1..4, {(1) = 0, (2) = 0, (3) = 0, (4) = 0}, datatype = integer[8]), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..8, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = 0, (2) = 0, (3) = 0, (4) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..4, {(1) = 84929.0, (2) = 299.0, (3) = 71411.0, (4) = 2304219.0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = -7.024349237099997, (2) = 15.35526343, (3) = -4.669572940000002, (4) = -7.2971383928999956}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..4, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = E(t), Y[2] = I(t), Y[3] = R(t), Y[4] = S(t)]`; YP[1] := -0.901230000000000e-3*Y[1]+0.100900000000000e-6*Y[4]*Y[2]; YP[2] := -.204255330000000*Y[2]+0.8999e-3*Y[1]; YP[3] := -0.901540000000000e-3*Y[3]+.1997*Y[2]; YP[4] := -0.133000000000000e-5*Y[4]-0.100900000000000e-6*Y[4]*Y[2]+0.90021e-3*Y[3]+.99879; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = E(t), Y[2] = I(t), Y[3] = R(t), Y[4] = S(t)]`; YP[1] := -0.901230000000000e-3*Y[1]+0.100900000000000e-6*Y[4]*Y[2]; YP[2] := -.204255330000000*Y[2]+0.8999e-3*Y[1]; YP[3] := -0.901540000000000e-3*Y[3]+.1997*Y[2]; YP[4] := -0.133000000000000e-5*Y[4]-0.100900000000000e-6*Y[4]*Y[2]+0.90021e-3*Y[3]+.99879; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 27 ) = (""), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0), ( 28 ) = (0)  ] ))  ] ); _y0 := Array(0..4, {(1) = 0., (2) = 84929., (3) = 299., (4) = 71411.}); _vmap := array( 1 .. 4, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3), ( 4 ) = (4)  ] ); _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); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `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 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 elif type(_xin, `=`) and lhs(_xin) = "setdatacallback" then if not type(rhs(_xin), 'nonegint') then error "data callback must be a nonnegative integer (address)" end if; _dtbl[1][28] := rhs(_xin) 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 _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; 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]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _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) = [t, E(t), I(t), R(t), S(t)], (4) = []}); _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 := 1; _ndsol := _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

 

 

 

 

 

 

Download odeProb.mw

using a simple rotation matrix is shown in the attached

  restart;
  with(plots):
  a := 7:
  b := a/2:
  rot:= Matrix(2,2, [ [cos(theta), -sin(theta)],
                      [sin(theta),  cos(theta)]
                  ]
            ):
  T:= rot.Vector(2, [x,y]):
  Ell:=x^2/a^2+y^2/b^2=1;

  doRot:= ang->expand(eval(Ell,eval( [x=T[1], y=T[2]], theta=ang))):
  e1:= doRot(Pi/4);
  e2:= doRot(-Pi/4);
  display( [ implicitplot(e1, x=-6..6, y=-6..6, color=blue),
             implicitplot(e2, x=-6..6, y=-6..6, color=blue)
           ]
         );

(1/49)*x^2+(4/49)*y^2 = 1

 

(5/98)*x^2+(3/49)*x*y+(5/98)*y^2 = 1

 

(5/98)*x^2-(3/49)*x*y+(5/98)*y^2 = 1

 

 

 

Download rotMat.mw

be using the geometry() package, as in the attached

  restart;
  with(geometry):
  a := 7:
  b := a/2:
  r := b/2:
  c := 2*sqrt(10)*r:
  circle(C1, [point(P1, [ c/2 - r, -c/2 + r]), r]):
  circle(C2, [point(P2, [-c/2 + r, -c/2 + r]), r]):
  circle(C3, [point(P3, [-c/2 + r,  c/2 - r]), r]):
  circle(C4, [point(P4, [ c/2 - r,  c/2 - r]), r]):
  ellipse(E, x^2/a^2 + y^2/b^2 = 1, [x,y]):
  rotation(e1, E, Pi/4, clockwise):
  rotation(e2, E, Pi/4, counterclockwise):
  Equation(e1);
  Equation(e2);
  draw( [ C1(color=cyan, filled=true),
          C2(color=cyan, filled=true),
          C3(color=cyan, filled=true),
          C4(color=cyan, filled=true),
          e1(color=blue),
          e2(color=blue)
        ],
        view=[-6..6, -6..6]
      );
  

(5/98)*y^2+(3/49)*x*y+(5/98)*x^2-1 = 0

 

(5/98)*x^2-(3/49)*x*y+(5/98)*y^2-1 = 0

 

 

 

Download geomProb.mw

but since no-one else is contributing, I'm going to do my bets with some (possibly incomplete?) guidelines

I have downloaded the DirectSearch package as directed in help text but the only result from issuing the command kernelopts(toolboxdir) is " ".

I don't think that this is a very good way to check whether or not you have *successfully* installed the DirectSearch() package. A better alternative would be

  with(LibraryTools):
  FindLibrary(DirectSearch);

If the DirectSearch() package has been installed then the result of the command  If this command givesFindLibrary() will give the location where the DirectSearch.mla repository is installed. If this command gives nothing in return, then Maple cannot find the required file in your current search path. Your current search path is determined by the output of the command libname -as in

libname;

This will return a list of directories where Maple will look for stuff (such as the DirectSearch package). I suppoes it is possible that yu have managed to install the DirectSearch package, somewhere weird, but assuming that you just followed default installation instructions, then I would expect the the DirectSearch package to be in one of two locations (on a Windows machine!)

  1. "C:Program Files\Maple 2022\lib\DirectSearch.mla", or
  2. C:\Users\yourUserName\maple\toolbox

I can only recommend that you have a good look into/around these locations.

##########################################
Another *quick and dirty* method of determing whether or not the DirectSeacrh package has been successfully installed is to use Maple's help facility - because it is a pretty good bet hta if the "help" for the DirectSearch package has been installed then it is a good bet that that the executable DirectSearch package has been installled! Steps are

  1. From a Maple worksheet use the menu entries Help->Maple Help
  2. In the Maple help browser near the top left, there is a "toggle" button labelled Products. I'm pretty sure that (by default) this will be set to 'Maple' - which means that you can only see help for "standard" Maple. However you can change htis to check the tickbox identified as 'User Help'. This will enable the "help" facility for any "user" packages you may have installed (such as DirectSearch)
  3. Having made sure that "user" help has been invoked (see item 2 above), type DirectSearch in the "Search" window. If you get  get a few pages reference in the left-hand pane, then the DirectSearch "help" is available - and by assumption, all DirectSearch commands are available

##########################################

Using Maple Cloud as means of exchanging files between computers. My first reaction is that  it seems like "overkill" (use email?) but it is possible, assuming you understand some limitations, about what you can subsequently do on these computers. This is not something I do often(/ever?) so I am working from the Maple help, part of which states

When you open a package workbook from the MapleCloud, you can use the package commands within that workbook.
However, if you want to be able to access those same package commands from within any Maple document, you must install the package workbook.

This suggests to me that yiu cannot modify or utilise (in a worksheet) anyhting from the Maple Cloud - you have to download and install it on the local machine. Haviing "installed" the relevant package, you can use standard commands from the LibraryTools() package to update the mla file any way you see fit. And then upload the results to Maple |Cloud again

maybe the attached will achieve what you want

  restart;
  padMatrix:= proc( M1, M2 )
                    uses LinearAlgebra:
                    if   op([1,2],M2) > op([1,2],M1)
                    then return `<|>`
                                ( Column(M1, 1..-1),
                                  Vector[column](op([1,1],M1))
                                  $
                                  op([1,2],M2)-op([1,2],M1)
                                );
                    elif op([1,2],M2) < op([1,2],M1)
                    then return `<|>`
                                ( Column(M2, 1..-1),
                                  Vector[column](op([1,1],M2))
                                  $ op([1,2],M1)-op([1,2],M2)
                                );
                    fi;
              end proc:
  A:=LinearAlgebra:-RandomMatrix(9,4):
  B:=LinearAlgebra:-RandomMatrix(6,7):
  padMatrix(A, B);
  padMatrix(B, A);

Matrix(9, 7, {(1, 1) = 25, (1, 2) = -22, (1, 3) = 57, (1, 4) = 27, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (2, 1) = 94, (2, 2) = 45, (2, 3) = 27, (2, 4) = 8, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (3, 1) = 12, (3, 2) = -81, (3, 3) = -93, (3, 4) = 69, (3, 5) = 0, (3, 6) = 0, (3, 7) = 0, (4, 1) = -2, (4, 2) = -38, (4, 3) = -76, (4, 4) = 99, (4, 5) = 0, (4, 6) = 0, (4, 7) = 0, (5, 1) = 50, (5, 2) = -18, (5, 3) = -72, (5, 4) = 29, (5, 5) = 0, (5, 6) = 0, (5, 7) = 0, (6, 1) = 10, (6, 2) = 87, (6, 3) = -2, (6, 4) = 44, (6, 5) = 0, (6, 6) = 0, (6, 7) = 0, (7, 1) = -16, (7, 2) = 33, (7, 3) = -32, (7, 4) = 92, (7, 5) = 0, (7, 6) = 0, (7, 7) = 0, (8, 1) = -9, (8, 2) = -98, (8, 3) = -74, (8, 4) = -31, (8, 5) = 0, (8, 6) = 0, (8, 7) = 0, (9, 1) = -50, (9, 2) = -77, (9, 3) = -4, (9, 4) = 67, (9, 5) = 0, (9, 6) = 0, (9, 7) = 0})

 

Matrix(%id = 36893488148081541356)

(1)

 

Download padMatrix.mw

using 2D input in Document Mode, is shown in the attached

dsolve(diff(y(x), x, x)+diff(y(x), x)+y(x) = 0)

y(x) = _C1*exp(-(1/2)*x)*sin((1/2)*3^(1/2)*x)+_C2*exp(-(1/2)*x)*cos((1/2)*3^(1/2)*x)

(1)

diff(%, x)

diff(y(x), x) = -(1/2)*_C1*exp(-(1/2)*x)*sin((1/2)*3^(1/2)*x)+(1/2)*_C1*exp(-(1/2)*x)*3^(1/2)*cos((1/2)*3^(1/2)*x)-(1/2)*_C2*exp(-(1/2)*x)*cos((1/2)*3^(1/2)*x)-(1/2)*_C2*exp(-(1/2)*x)*3^(1/2)*sin((1/2)*3^(1/2)*x)

(2)

simplify(%)

diff(y(x), x) = (1/2)*exp(-(1/2)*x)*((3^(1/2)*_C1-_C2)*cos((1/2)*3^(1/2)*x)-(3^(1/2)*_C2+_C1)*sin((1/2)*3^(1/2)*x))

(3)

NULL

Download doDiff.mw

just a faiure on the OP's part to understand how Maple *categorizes* an expression, and therefore what will be returned when using convert(expression, list).

The attached contains examples (with comments) for clarification.

  restart:

#
# Certainly not a bug, just OP's failure to understand
# how Maple categorizes expressions
#
# For example, what type is the following expression?
#
  expr:=U(xi)^2*y(xi)^5;
#
# Hmmmm - multiplicative
#
  whattype(expr);
#
# So what are the terms in the multiplicative expression?
#
  convert(expr,list);

U(xi)^2*y(xi)^5

 

`*`

 

[U(xi)^2, y(xi)^5]

(1)

#
# Let's make it more complicated. How does Maple
# categorize the following expression
#
  expr:=U(xi)^2*y(xi)^5*sin(xi);
#
# Hmmmm - multiplicative
#
  whattype(expr);
#
# So what are the terms in the multiplicative expression?
#
  convert(expr,list);

U(xi)^2*y(xi)^5*sin(xi)

 

`*`

 

[U(xi)^2, y(xi)^5, sin(xi)]

(2)

#
# Let's make it *even* more complicated. How does Maple
# categorize the following expression
#
  expr:=U(xi)^2*y(xi)^5*sin(xi)+diff(y(xi), xi);
#
# Hmmmm - additive
#
  whattype(expr);
#
# So what are the terms in the additive expression?
#
  convert(expr,list);

U(xi)^2*y(xi)^5*sin(xi)+diff(y(xi), xi)

 

`+`

 

[U(xi)^2*y(xi)^5*sin(xi), diff(y(xi), xi)]

(3)

#
# Let's simplify a little How does Maple
# categorize the following expression
#
  expr:=U(xi)^2;
#
# Hmmmm - power
#
  whattype(expr);
#
# So what are the terms in the power expression?
#
  convert(expr,list);

U(xi)^2

 

`^`

 

[U(xi), 2]

(4)

 

Download exprtype.mw

square brackets (ie '[]') for simple grouping of terms in an algebraic expression.

See the attached

restartNULL

Lagrangian

 

Leq := (1/2)*R^2*(-cosh(rho1(tau)^2)*(diff(t(tau), tau))^2+(diff(rho1(tau), tau))^2+sinh(rho1(tau)^2)*(diff(theta1(tau), tau))^2)+(1/2)*R^2*(-cosh(rho2(tau)^2)*(diff(t(tau), tau))^2+(diff(rho2(tau), tau))^2+sinh(rho2(tau)^2)*(diff(theta2(tau), tau))^2)-(1/2)*k*rho1(tau)^2-(1/2)*k*rho2(tau)^2-(1/2*(tanh(rho2(tau)-rho1(tau))+1))*(rho2(tau)-rho1(tau))^2; L := subs({diff(rho1(tau), tau) = var4, diff(rho2(tau), tau) = var6, diff(t(tau), tau) = var2, diff(theta1(tau), tau) = var8, diff(theta2(tau), tau) = var10, rho1(tau) = var3, rho2(tau) = var5, t(tau) = var1, theta1(tau) = var7, theta2(tau) = var9}, Leq)

(1/2)*R^2*(-cosh(rho1(tau)^2)*(diff(t(tau), tau))^2+(diff(rho1(tau), tau))^2+sinh(rho1(tau)^2)*(diff(theta1(tau), tau))^2)+(1/2)*R^2*(-cosh(rho2(tau)^2)*(diff(t(tau), tau))^2+(diff(rho2(tau), tau))^2+sinh(rho2(tau)^2)*(diff(theta2(tau), tau))^2)-(1/2)*k*rho1(tau)^2-(1/2)*k*rho2(tau)^2-(1/2)*(-tanh(-rho2(tau)+rho1(tau))+1)*(rho2(tau)-rho1(tau))^2

(1.1)

NULL

Time Equation

 

epr11 := diff(L, var2); epr12 := diff(L, var1); epr13 := subs({var1 = t(tau), var10 = diff(theta2(tau), tau), var2 = diff(t(tau), tau), var3 = rho1(tau), var4 = diff(rho1(tau), tau), var5 = rho2(tau), var6 = diff(rho2(tau), tau), var7 = theta1(tau), var8 = diff(theta1(tau), tau), var9 = theta2(tau)}, epr11); epr14 := subs({var1 = t(tau), var10 = diff(theta2(tau), tau), var2 = diff(t(tau), tau), var3 = rho1(tau), var4 = diff(rho1(tau), tau), var5 = rho2(tau), var6 = diff(rho2(tau), tau), var7 = theta1(tau), var8 = diff(theta1(tau), tau), var9 = theta2(tau)}, epr12); epr15 := diff(epr13, tau); teq := epr15-epr14 = 0

-2*R^2*rho1(tau)*(diff(rho1(tau), tau))*sinh(rho1(tau)^2)*(diff(t(tau), tau))-R^2*cosh(rho1(tau)^2)*(diff(diff(t(tau), tau), tau))-2*R^2*rho2(tau)*(diff(rho2(tau), tau))*sinh(rho2(tau)^2)*(diff(t(tau), tau))-R^2*cosh(rho2(tau)^2)*(diff(diff(t(tau), tau), tau)) = 0

(2.1)

NULL

NULL

Download badBracket.mw

Form the help page for the ShowSolution() command (emphasis added)

The ShowSolution command is used to show the solution steps for a Calculus1 problem, that is, a limit, differentiation or integration problem such as can be expected to be encountered in a single-variable calculus course.

 

use square brackets (ie '[]') for grouping terms in a simple algebraic expression. Square brackets in Maple are used to construct indexable quantities, such as lists.

Just changing these square brackets, code runs - see the attached

restart

with(Optimization); with(plots)

d := .75

.75

(1)

f__s := 200*10^3

200000

(2)

obj := k__v*(d^(1-alpha)+(1-d)^(1-alpha))*2^alpha*f__s^alpha*B__peak^beta/(4*(2*Pi)^(alpha-1)*(.2761+1.7061/(alpha+1.354)))

(1/4)*k__v*(.75^(1-alpha)+.25^(1-alpha))*2^alpha*200000^alpha*B__peak^beta/((2*Pi)^(alpha-1)*(.2761+1.7061/(alpha+1.354)))

(3)

cnsts := [.5 <= alpha, alpha <= 3, .1 <= beta, beta <= 10, .1 <= k__v, k__v <= 10, 0.1e-1 <= B__peak, B__peak <= .3]

[.5 <= alpha, alpha <= 3, .1 <= beta, beta <= 10, .1 <= k__v, k__v <= 10, 0.1e-1 <= B__peak, B__peak <= .3]

(4)

NLPSolve(obj, alpha = .5 .. 3, beta = .1 .. 10, k__v = .1 .. 10, B__peak = 0.1e-1 .. .3, initialpoint = [alpha = .5, beta = .1, k__v = .1, B__peak = 0.1e-1])

[0.967042530117581976e-14, [B__peak = HFloat(0.01), alpha = HFloat(0.4999999998854566), beta = HFloat(8.47772257127817), k__v = HFloat(1.9285093099163284)]]

(5)

``

Download optProb2.mw

 

the attached solves both of your problems


 

  restart:
  with(LinearAlgebra):
  uniqRow:= mat-> local i,j;
                  `if`( member
                        ( true,
                          { seq
                            ( seq
                              ( Equal
                                ( mat[j,..],
                                  mat[i,..]
                                ),
                                i=j+1..op([1,1],mat)
                              ),
                              j=1..op([1,1],mat)
                            )
                          }
                        ),
                        false,
                        true
                      ):
#
# The odds of getting two identical rows in
# a random matrix is infinitesimal
#
  M:=RandomMatrix(10,10):
  uniqRow(M);
#
# Force two rows to be the same and check again
#
  M[5,..]:=M[1,..]:
  uniqRow(M);

true

 

false

(1)

  A:=Matrix(3, 3, [[1, 2, 3], [3, 2, 1], [4, 6, 5]]):
  B:=Matrix(3, 3, [[3, 2, 1], [1, 2, 3], [4, 6, 5]]):
  C:=Matrix(3, 3, [[3, 2, 1], [1, 2, 3], [4, 5, 6]]):
  isPerm:= (mat1, mat2)-> local i,j;
                          `if`
                          ( member
                            ( true,
                              { seq
                                ( `if`
                                  ( Equal
                                    ( mat1,
                                      `<,>`( seq(mat2[i,..], i in j))
                                    ),
                                    true,
                                    false
                                  ),
                                  j in combinat:-permute([$1..op([1,1],mat1)])
                                )
                              }
                            ),
                            true,
                            false
                          ):
  isPerm(A,B);
  isPerm(A,C);

true

 

false

(2)

 


 

Download matManip.mw

The function definition

f:=(r,theta)->I*sinh(theta-Pi/6)+cos(Pi/4)=r;

could be (trivially) rewitten as

f:=(r,theta)->I*sinh(theta-Pi/6)=r-cos(Pi/4);

If both 'theta' and 'r' are real, then the left hand side of this latter equation is purely imaginary, and the right-hand side is purely real. So under what circumstances could they possibly be equal? Only if both sides are zero, which requires theta=Pi/6, r=cos(Pi/4)=sqrt(2)/2.

In other words you original equation is only satisfied at a single point. Now what exactly do you want to plot?

repeated application of Dijkstra's algorithm - as in the attached?

  restart;
  with(GraphTheory):
  doDijk:= proc( GG::Graph, start )
                 local j;
                 uses GraphTheory, ListTools:
                 return seq
                        ( Reverse
                          ( DijkstrasAlgorithm
                            ( G2,
                              start,
                              j
                            )
                          ),
                          j in `minus`
                               ( convert
                                 ( Vertices(G2),
                                   set
                                 ),
                                 {start}
                               )
                         );
          end proc:

 G2:=Graph( [$1..5], {{2,3}, {2,4}, {3,4}, {3,5}}):
 doDijk( G2, 1);
 doDijk( G2, 2);
 doDijk( G2, 3);
 doDijk( G2, 4);
 doDijk( G2, 5);

[infinity, []], [infinity, []], [infinity, []], [infinity, []]

 

[infinity, []], [1, [2, 3]], [1, [2, 4]], [2, [2, 3, 5]]

 

[infinity, []], [1, [3, 2]], [1, [3, 4]], [1, [3, 5]]

 

[infinity, []], [1, [4, 2]], [1, [4, 3]], [2, [4, 3, 5]]

 

[infinity, []], [2, [5, 3, 2]], [1, [5, 3]], [2, [5, 3, 4]]

(1)

 

Download dijk.mw

 

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