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How to plot results from ode-system?...

Asked by: Alfred_F 380
ode dsolve numeric
September 18 2025
1  9

In the attached file "test," a system of ordinary differential equations is solved. I was able to create the (r, phi) plot. But how are the (t, r(t)) and (t, phi(t)) plots executed with the calculation results?
What does "range" mean in the (r, phi) plot command? Is there a more precise numerical method than the Runge-Kutta method used in "test" (perhaps a finer t-division of the axis?)?

> 

restart
> 

eq1 := diff(r(t), t)+1+cos(`ϕ`(t)) = 0; eq2 := diff(`ϕ`(t), t)+1-sin(`ϕ`(t))/r(t) = 0

diff(r(t), t)+1+cos(varphi(t)) = 0

  

diff(varphi(t), t)+1-sin(varphi(t))/r(t) = 0

 (1)
> 

``
> 

ics := r(0) = 2.0, `ϕ`(0) = Pi/(1.5)

r(0) = 2.0, varphi(0) = 2.094395103

 (2)
> 

soln := dsolve({eq1, eq2, ics}, {r(t), `ϕ`(t)}, numeric, start = 0, range = 0 .. 3)

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, ( "right" ) = 3., ( "left" ) = 0. ] ) _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) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 1, (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) = 3.0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .19535812688284548, (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..2, {(1) = 2.0, (2) = 2.094395103}, 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..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .22047804756371955, (2) = 2.873107829856247}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = -0.3386381880498868e-1, (2) = .17766918858144054}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = -0.3386381880498868e-1, (1, 2) = -0.377585836518608e-1, (1, 3) = -0.3708633582928045e-1, (1, 4) = -0.34124098676952874e-1, (1, 5) = -0.3363280624230314e-1, (1, 6) = -0.3641604213065852e-1, (2, 1) = .17766918858144054, (2, 2) = .22761845480999643, (2, 3) = .2192661580732893, (2, 4) = .18085452152511095, (2, 5) = .17405774363177118, (2, 6) = .2108205409669659}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .22040658040387567, (2) = 2.873512341267432}, datatype = float[8], order = C_order), Array(1..2, {(1) = .21910457742849349, (2) = 2.8806072512024623}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0.17347438180381758e-7, (2) = 0.10386005389051434e-7}, datatype = float[8], order = C_order), Array(1..2, {(1) = .22270057794927467, (2) = 2.8597843388449604}, datatype = float[8], order = C_order), Array(1..2, {(1) = -0.3569387106220412e-1, (2) = .20146120823190228}, 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..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = 2.0, (2) = 2.094395103}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = -.4999999994744918, (2) = -.5669872982594819}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = varphi(t)]`; YP[1] := -1-cos(Y[2]); YP[2] := -1+sin(Y[2])/Y[1]; 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, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = varphi(t)]`; YP[1] := -1-cos(Y[2]); YP[2] := -1+sin(Y[2])/Y[1]; 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) ] )), ( 3 ) = (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) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 1, (9) = 0, (10) = 1, (11) = 85, (12) = 85, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 155, (19) = 30000, (20) = 5, (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) = 3.0, (2) = 0.10e-5, (3) = .10304186874784538, (4) = 0.500001e-14, (5) = .0, (6) = .19535812688284548, (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..2, {(1) = 2.0, (2) = 2.094395103}, 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..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .22047804756371955, (2) = 2.873107829856247}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = -0.3386381880498868e-1, (2) = .17766918858144054}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = -0.3386381880498868e-1, (1, 2) = -0.377585836518608e-1, (1, 3) = -0.3708633582928045e-1, (1, 4) = -0.34124098676952874e-1, (1, 5) = -0.3363280624230314e-1, (1, 6) = -0.3641604213065852e-1, (2, 1) = .17766918858144054, (2, 2) = .22761845480999643, (2, 3) = .2192661580732893, (2, 4) = .18085452152511095, (2, 5) = .17405774363177118, (2, 6) = .2108205409669659}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .22040658040387567, (2) = 2.873512341267432}, datatype = float[8], order = C_order), Array(1..2, {(1) = .21910457742849349, (2) = 2.8806072512024623}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0.17347438180381758e-7, (2) = 0.10386005389051434e-7}, datatype = float[8], order = C_order), Array(1..2, {(1) = .22270057794927467, (2) = 2.8597843388449604}, datatype = float[8], order = C_order), Array(1..2, {(1) = -0.3569387106220412e-1, (2) = .20146120823190228}, 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..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = .22270057794927467, (2) = 2.8597843388449604}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = -0.3386381880498868e-1, (2) = .17766918858144054}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = 2.938831384122585, (1, 2) = .22270057794927467, (2, 0) = .22270057794927467, (2, 1) = 2.8597843388449604, (2, 2) = 2.9634858657986514, (3, 0) = 2.9634858657986514, (3, 1) = .221748342025735, (3, 2) = 2.8656651287650265, (4, 0) = 2.8656651287650265, (4, 1) = 2.988140347474718, (4, 2) = .22083401075704298, (5, 0) = .22083401075704298, (5, 1) = 2.871070168038091, (5, 2) = 3.012794829150785, (6, 0) = 3.012794829150785, (6, 1) = .21995387131675959, (6, 2) = 2.876038628847415}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = varphi(t)]`; YP[1] := -1-cos(Y[2]); YP[2] := -1+sin(Y[2])/Y[1]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (Array(1..85, 0..2, {(1, 1) = .0, (1, 2) = 2.0, (2, 0) = 2.0, (2, 1) = 2.094395103, (2, 2) = 0.4883953172071137e-1, (3, 0) = 0.4883953172071137e-1, (3, 1) = 1.9749957636720312, (3, 2) = 2.0670025471235336, (4, 0) = 2.0670025471235336, (4, 1) = 0.9767906344142274e-1, (4, 2) = 1.9488280959770066, (5, 0) = 1.9488280959770066, (5, 1) = 2.04021084709857, (5, 2) = .1465185951621341, (6, 0) = .1465185951621341, (6, 1) = 1.9215068608366304, (6, 2) = 2.0140263347936975, (7, 0) = 2.0140263347936975, (7, 1) = .19535812688284548, (7, 2) = 1.893044054829213, (8, 0) = 1.893044054829213, (8, 1) = 1.988457425914424, (8, 2) = .280900476384457, (9, 0) = .280900476384457, (9, 1) = 1.8404843242365583, (9, 2) = 1.9451906076368206, (10, 0) = 1.9451906076368206, (10, 1) = .36644282588606847, (10, 2) = 1.7845598519772645, (11, 0) = 1.7845598519772645, (11, 1) = 1.9039191638716142, (11, 2) = .4519851753876799, (12, 0) = .4519851753876799, (12, 1) = 1.7253847367916073, (12, 2) = 1.8647415453538858, (13, 0) = 1.8647415453538858, (13, 1) = .5375275248892915, (13, 2) = 1.6630931456083156, (14, 0) = 1.6630931456083156, (14, 1) = 1.827785275008053, (14, 2) = .616075451639432, (15, 0) = .616075451639432, (15, 1) = 1.603283300376125, (15, 2) = 1.7959429678186245, (16, 0) = 1.7959429678186245, (16, 1) = .6946233783895726, (16, 2) = 1.5411171896322202, (17, 0) = 1.5411171896322202, (17, 1) = 1.7662563673049378, (17, 2) = .7731713051397131, (18, 0) = .7731713051397131, (18, 1) = 1.4767537695478887, (18, 2) = 1.7389080337794152, (19, 0) = 1.7389080337794152, (19, 1) = .8517192318898537, (19, 2) = 1.4103708331171143, (20, 0) = 1.4103708331171143, (20, 1) = 1.7141191124530415, (20, 2) = .9177358538213208, (21, 0) = .9177358538213208, (21, 1) = 1.3531616403350515, (21, 2) = 1.6954560815446786, (22, 0) = 1.6954560815446786, (22, 1) = .983752475752788, (22, 2) = 1.2948000182859192, (23, 0) = 1.2948000182859192, (23, 1) = 1.6789696037423298, (23, 2) = 1.0497690976842553, (24, 0) = 1.0497690976842553, (24, 1) = 1.235434387752726, (24, 2) = 1.664875276796809, (25, 0) = 1.664875276796809, (25, 1) = 1.1157857196157224, (25, 2) = 1.1752283323792916, (26, 0) = 1.1752283323792916, (26, 1) = 1.6534251811289784, (26, 2) = 1.17181879714737, (27, 0) = 1.17181879714737, (27, 1) = 1.1236019582656207, (27, 2) = 1.6459973906746916, (28, 0) = 1.6459973906746916, (28, 1) = 1.2278518746790175, (28, 2) = 1.0716241501398995, (29, 0) = 1.0716241501398995, (29, 1) = 1.640888665777501, (29, 2) = 1.2838849522106648, (30, 0) = 1.2838849522106648, (30, 1) = 1.0194315393350855, (30, 2) = 1.6383329167946707, (31, 0) = 1.6383329167946707, (31, 1) = 1.3399180297423123, (31, 2) = .967173884862351, (32, 0) = .967173884862351, (32, 1) = 1.63859579173605, (32, 2) = 1.3895931786245055, (33, 0) = 1.3895931786245055, (33, 1) = .9209233700331131, (33, 2) = 1.6414255181271569, (34, 0) = 1.6414255181271569, (34, 1) = 1.4392683275066989, (34, 2) = .8748784230151254, (35, 0) = .8748784230151254, (35, 1) = 1.646939265151252, (35, 2) = 1.4889434763888922, (36, 0) = 1.4889434763888922, (36, 1) = .8291788145340493, (36, 2) = 1.6553957182885823, (37, 0) = 1.6553957182885823, (37, 1) = 1.5386186252710854, (37, 2) = .7839766015575426, (38, 0) = .7839766015575426, (38, 1) = 1.667077404650224, (38, 2) = 1.5687462276863915, (39, 0) = 1.5687462276863915, (39, 1) = .7568735457220451, (39, 2) = 1.67586213435023, (40, 0) = 1.67586213435023, (40, 1) = 1.5988738301016974, (40, 2) = .7300540461536099, (41, 0) = .7300540461536099, (41, 1) = 1.6860182095468146, (41, 2) = 1.6290014325170035, (42, 0) = 1.6290014325170035, (42, 1) = .7035599816612077, (42, 2) = 1.697620293665825, (43, 0) = 1.697620293665825, (43, 1) = 1.6591290349323096, (43, 2) = .6774351893350706, (44, 0) = .6774351893350706, (44, 1) = 1.7107443248264431, (44, 2) = 1.6892566373476157, (45, 0) = 1.6892566373476157, (45, 1) = .6517254613446625, (45, 2) = 1.7254668242982154, (46, 0) = 1.7254668242982154, (46, 1) = 1.7193842397629218, (46, 2) = .6264787480696691, (47, 0) = .6264787480696691, (47, 1) = 1.7418650301879928, (47, 2) = 1.7495118421782276, (48, 0) = 1.7495118421782276, (48, 1) = .6017446545171511, (48, 2) = 1.7600130123373363, (49, 0) = 1.7600130123373363, (49, 1) = 1.7796394445935337, (49, 2) = .5775742015482379, (50, 0) = .5775742015482379, (50, 1) = 1.7799796411830302, (50, 2) = 1.8061947635665625, (51, 0) = 1.8061947635665625, (51, 1) = .5567786183444859, (51, 2) = 1.7991366433344658, (52, 0) = 1.7991366433344658, (52, 1) = 1.832750082539591, (52, 2) = .5364978626253581, (53, 0) = .5364978626253581, (53, 1) = 1.8197927195419688, (53, 2) = 1.8593054015126196, (54, 0) = 1.8593054015126196, (54, 1) = .5167682561523157, (54, 2) = 1.8419764809526942, (55, 0) = 1.8419764809526942, (55, 1) = 1.8858607204856483, (55, 2) = .49762581223390895, (56, 0) = .49762581223390895, (56, 1) = 1.8657055776358582, (56, 2) = 1.9097149107830291, (57, 0) = 1.9097149107830291, (57, 1) = .4809602717718633, (57, 2) = 1.8883435286863, (58, 0) = 1.8883435286863, (58, 1) = 1.93356910108041, (58, 2) = .4648218596764317, (59, 0) = .4648218596764317, (59, 1) = 1.9122277759833848, (59, 2) = 1.9574232913777907, (60, 0) = 1.9574232913777907, (60, 1) = .4492340389199136, (60, 2) = 1.9373409026689865, (61, 0) = 1.9373409026689865, (61, 1) = 1.9812774816751715, (61, 2) = .4342187292534649, (62, 0) = .4342187292534649, (62, 1) = 1.9636521189398293, (62, 2) = 2.004263852787264, (63, 0) = 2.004263852787264, (63, 1) = .4203101092586252, (63, 2) = 1.990098117377601, (64, 0) = 1.990098117377601, (64, 1) = 2.027250223899356, (64, 2) = .4069674582769166, (65, 0) = .4069674582769166, (65, 1) = 2.0175622542309486, (65, 2) = 2.050236595011448, (66, 0) = 2.050236595011448, (66, 1) = .3942039633751688, (66, 2) = 2.0459772536535046, (67, 0) = 2.0459772536535046, (67, 1) = 2.07322296612354, (67, 2) = .38203013565972604, (68, 0) = .38203013565972604, (68, 1) = 2.075261255402451, (68, 2) = 2.098058121270015, (69, 0) = 2.098058121270015, (69, 1) = .3695485759576045, (69, 2) = 2.107766300909261, (70, 0) = 2.107766300909261, (70, 1) = 2.1228932764164896, (70, 2) = .3577689175737098, (71, 0) = .3577689175737098, (71, 1) = 2.141033682628699, (71, 2) = 2.1477284315629643, (72, 0) = 2.1477284315629643, (72, 1) = .3466910657927155, (72, 2) = 2.174907057374285, (73, 0) = 2.174907057374285, (73, 1) = 2.1725635867094395, (73, 2) = .33631017357287435, (74, 0) = .33631017357287435, (74, 1) = 2.20921581927419, (74, 2) = 2.196849383761043, (75, 0) = 2.196849383761043, (75, 1) = .3268235884660522, (75, 2) = 2.2430115231100825, (76, 0) = 2.2430115231100825, (76, 1) = 2.221135180812647, (76, 2) = .3179799207626713, (77, 0) = .3179799207626713, (77, 1) = 2.276869570385799, (77, 2) = 2.2454209778642507, (78, 0) = 2.2454209778642507, (78, 1) = .309760915843174, (78, 2) = 2.3106091885511706, (79, 0) = 2.3106091885511706, (79, 1) = 2.2697067749158544, (79, 2) = .3021447460238304, (80, 0) = .3021447460238304, (80, 1) = 2.3440521970551167, (80, 2) = 2.2915527181738296, (81, 0) = 2.2915527181738296, (81, 1) = .2957879507637747, (81, 2) = 2.373738294552842, (82, 0) = 2.373738294552842, (82, 1) = 2.3133986614318047, (82, 2) = .28987791521959144, (83, 0) = .28987791521959144, (83, 1) = 2.402922916310279, (83, 2) = 2.3352446046897803, (84, 0) = 2.3352446046897803, (84, 1) = .28439275840127815, (84, 2) = 2.4314945706700617, (85, 0) = 2.4314945706700617, (85, 1) = 2.3570905479477555, (85, 2) = .27930964339102904}, datatype = float[8], order = C_order)), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = varphi(t)]`; YP[1] := -1-cos(Y[2]); YP[2] := -1+sin(Y[2])/Y[1]; 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) ] )), ( 4 ) = (3) ] ); _y0 := Array(0..2, {(1) = 0., (2) = 2.0}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _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 <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> 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 <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> 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, r(t), varphi(t)], (4) = []}); _vars := _dat[3]; _pars := map(lhs, _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

 (3)
> 

plots[odeplot](soln, [r(t), `&varphi;`(t)])

 

Download test.mw

How in combined plot make the movement in both li...

Asked by: salim-barzani 1635
contourplot plotting duplicate-question
September 14 2025
1  4

In this kind of contour plot i have two line but when i change time variable t just contour of one line wil move the other is not do any movement and is stop how i can  make the second plot one second line move too? also there is any way for ploting this kind any other option?

line-2-done.mw

How test the function with a different kind for t...

Asked by: salim-barzani 1635
ode differential-equations odetest
September 12 2025
1  8

i did for two of them base on the information but one of them is not make my odetest to be zero? where is problem

test.mw

How perform quantum field theory (QFT) integrals?...

Asked by: nmacsai 265
integration physics kronecker
September 09 2025
1  0

It is possible to perform the simplest QFT calculations with second quantization, in Maple? Bosons in a box. See attached example. bosons_in_a_box.mw

Sure any general purpose programming language is capable of performing this task with enough effort. What I am interested in is if the physics tools has a standard way of dealing with these calculations. The general impedement when attempting the calculation is that integrations are perfomed by replacements with delta functions or kronecker delta functions, and its not clear how to force the Maple Physics package to recognize this or if that's possible. Part of the problem is that integrations in maple are defined in one dimension at a time where as in QFT the integration element is almost always atleast three dimensional, d^3x or dxdydzy, the later of which can get extremely cumbersome with even a small number of fields under consideration. I don't find much of what I am refering to mentioned in the help pages and I doubt these types of QFT calculations are possible to perform in Maple without addressing these issues.

bosons_in_a_box.mw

How fix the issue of this plotting?...

Asked by: Math-dashti 115
explore dsolve numeric
September 08 2025
2  6

i don't  know  why my graph make a problem and what is issue i did plot  but this time make issue for me which i don't know where is problem there is anyone which can help and even modified the plot?

explore-chaotic.mw

How get unknown after substitution by linearizat...

Asked by: salim-barzani 1635
substitution differential-equations
September 05 2025
0  9

i did substitution but my result is so different from the author i think he just take the linear term of theta but i didn't do that so how take just linear term of that function and find unknwon , and how afeter replacing eq(12) inside eq(11) we can remove thus exponential and find w? also i think author did a mistake which the equation 12 is theta(x,t) not Q(x,t)

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet)
> 

declare(u(x, t), quiet); declare(U(xi), quiet); declare(V(xi), quiet); declare(theta(x, t), quiet)
> 

pde := diff(u(x, t), `$`(t, 2))-s^2*(diff(u(x, t), `$`(x, 2)))+(2*I)*(diff(u(x, t)*U^2, t))-(2*I)*alpha*s*(diff(u(x, t)*U^2, t))+I*(diff(u(x, t), `$`(x, 2), t))-I*beta*s*(diff(u(x, t), `$`(x, 3)))

diff(diff(u(x, t), t), t)-s^2*(diff(diff(u(x, t), x), x))+(2*I)*(diff(u(x, t), t))*U^2-(2*I)*alpha*s*(diff(u(x, t), t))*U^2+I*(diff(diff(diff(u(x, t), t), x), x))-I*beta*s*(diff(diff(diff(u(x, t), x), x), x))

 (1)
> 

T := u(x, t) = (sqrt(Q)+theta(x, t))*exp(I*(Q^2*epsilon*gamma+Q*q)*t); T1 := U = sqrt(Q)+theta(x, t)

u(x, t) = (Q^(1/2)+theta(x, t))*exp(I*(Q^2*epsilon*gamma+Q*q)*t)

  

U = Q^(1/2)+theta(x, t)

 (2)
> 

P := collect(eval(subs({T, T1}, pde)), exp)/exp(I*(Q^2*gamma*`&epsilon;`+Q*q)*t)

diff(diff(theta(x, t), t), t)+(2*I)*(diff(theta(x, t), t))*(Q^2*epsilon*gamma+Q*q)-(Q^(1/2)+theta(x, t))*(Q^2*epsilon*gamma+Q*q)^2-s^2*(diff(diff(theta(x, t), x), x))+(2*I)*(diff(theta(x, t), t)+I*(Q^(1/2)+theta(x, t))*(Q^2*epsilon*gamma+Q*q))*(Q^(1/2)+theta(x, t))^2-(2*I)*alpha*s*(diff(theta(x, t), t)+I*(Q^(1/2)+theta(x, t))*(Q^2*epsilon*gamma+Q*q))*(Q^(1/2)+theta(x, t))^2+I*(diff(diff(diff(theta(x, t), t), x), x)+I*(diff(diff(theta(x, t), x), x))*(Q^2*epsilon*gamma+Q*q))-I*beta*s*(diff(diff(diff(theta(x, t), x), x), x))

 (3)
> 

 

> 

TT := Q = alpha[1]*exp(I*(k*x-t*w))+alpha[2]*exp(-I*(k*x-t*w))

Q = alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w))

 (4)
> 

S := eval(subs(TT, P))

diff(diff(theta(x, t), t), t)+(2*I)*(diff(theta(x, t), t))*(gamma*epsilon*(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^2+(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))*q)-((alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^(1/2)+theta(x, t))*(gamma*epsilon*(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^2+(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))*q)^2-s^2*(diff(diff(theta(x, t), x), x))+(2*I)*(diff(theta(x, t), t)+I*((alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^(1/2)+theta(x, t))*(gamma*epsilon*(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^2+(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))*q))*((alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^(1/2)+theta(x, t))^2-(2*I)*alpha*s*(diff(theta(x, t), t)+I*((alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^(1/2)+theta(x, t))*(gamma*epsilon*(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^2+(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))*q))*((alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^(1/2)+theta(x, t))^2+I*(diff(diff(diff(theta(x, t), t), x), x)+I*(diff(diff(theta(x, t), x), x))*(gamma*epsilon*(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))^2+(alpha[1]*exp((k*x-t*w)*I)+alpha[2]*exp(-I*(k*x-t*w)))*q))-I*beta*s*(diff(diff(diff(theta(x, t), x), x), x))

 (5)

Download steps.mw

or for this equation 

steps-2.mw

what is issue in thus odes which not make my funct...

Asked by: salim-barzani 1635
August 30 2025
1  2

I test a lot of them but some of them make a problem i  don't know i am do it in wrong way or the author did wrong i need verifying thus solution of odes specially in case 4 when we have not equal sign how use that?

and case 5 is Weierstrass elliptic function which i don't know how set up and use i think is a on kinf of odes but why they use that sign for this function?

ode-17.mw

there’s a way to link each curve in the animation ...

Asked by: jalal 445
animation textplot plotting
August 26 2025
0  6

Hi,

I’m having fun animating a beautiful geometric shape starting from a few trigonometric functions. I’m wondering if there’s a way to link each curve in the animation to its name.

> 

restart
> 

plots:-animatecurve([sin(x), sin(x)^2, sin(x)^3, sin(x)^4, sin(x)^5, sin(x)^6, surd(sin(x), 2), surd(sin(x), 3), surd(sin(x), 4), surd(sin(x), 5), surd(sin(x), 6)], x = 0 .. Pi, thickness = 2.5, background = "AliceBlue", labels = ["", ""], size = [800, 800])

 
> 

plot([sin(x), sin(x)^2, sin(x)^3, sin(x)^4, sin(x)^5, sin(x)^6, surd(sin(x), 2), surd(sin(x), 3), surd(sin(x), 4), surd(sin(x), 5), surd(sin(x), 6)], x = 0 .. Pi, thickness = 2.5)

 

NULL

Download Animation_Trigo.mw

Is it possible to resolve the cursor-jumping issue...

Asked by: ider 90
2dmath gui editing
August 26 2025
1  1

I recently upgraded from Maple 23 to Maple 24. While many display issues have been resolved, I’ve encountered a new real problem: when entering operations or a factorial in a denominator or exponent, the cursor unexpectedly jumps to the inline position. This forces me to manually reposition the cursor using backspace plus left-arrow keys, and forgetting to do so can lead to errors. I have a perpetual Maple license through my university and haven’t purchased maintenance this time. Is there any way to fix or work around this cursor-jumping issue in Maple 24 without purchasing a new license?

How find this 3-steps for Schrodinger equations ?...

Asked by: salim-barzani 1635
substitution pde differential-equations
August 26 2025
3  8

Already  by Help of my favorite Dr.David he did find the thus three step for non schrodinger equation but in here i got some issue of coding which is so different from before, is about transformation of pdes to two parts od real and imaginary part and then substitution our function the functions is clear but combining them and findinf leading exponent and resonance point and finding function in step 3 in different and jsut the function is different with eperate the real and imaginary part for finding step one ...

note:=q=u*exp(#) then |q|=u

schrodinger-test.mw

paper-1

paper-2

Conversion of a term...

Asked by: Alfred_F 380
radical trig
August 25 2025
2  3

In the attached file, the trigonometric term (2, term) is transformed into a term (3, term1) consisting of radicals. Is there a Maple procedure that can be used to reverse this process? Given an algebraic term (e.g., consisting of radicals, powers, etc.), under what conditions can it be transformed into a trigonometric form (not a Fourier series) in the sense of (3) according to (2)?test.mw

> 

 interface(version);

`Standard Worksheet Interface, Maple 2024.2, Windows 11, October 29 2024 Build ID 1872373`

 (1)

restart

term := 2*cos(5*arcsin((1/2)*x))

2*cos(5*arcsin((1/2)*x))

 (2)
> 

term1 := expand(term)

-3*(-x^2+4)^(1/2)*x^2+(-x^2+4)^(1/2)+(-x^2+4)^(1/2)*x^4

 (3)
> 

convert(term1, trig)

-3*(-x^2+4)^(1/2)*x^2+(-x^2+4)^(1/2)+(-x^2+4)^(1/2)*x^4

 (4)
> 

simplify(term1, trig)

(-x^2+4)^(1/2)*(x^4-3*x^2+1)

 (5)

solve(term1 = sqrt(2), x)

(1/2)*(8-2*(10+2*5^(1/2))^(1/2))^(1/2), (1/2)*(8+2*(10-2*5^(1/2))^(1/2))^(1/2), (1/2)*(8+2*(10+2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8-2*(10+2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8+2*(10-2*5^(1/2))^(1/2))^(1/2), -(1/2)*(8+2*(10+2*5^(1/2))^(1/2))^(1/2)

 (6)
> 

evalf(solve(term1 = sqrt(2), x))

.3128689302, 1.782013048, 1.975376681, -.3128689302, -1.782013048, -1.975376681

 (7)

plot(term, x = -2.5 .. 2.5)

  

plot(term1, x = -2.5 .. 2.5)

  

NULL

Download test.mw

How find leading order & resonance point ?...

Asked by: salim-barzani 1635
parameters pde differential-equations
August 21 2025
2  37

is been a while i work on a test still i am study and there is a lot paper remain and is so important in PDEs, a lot paper explain in 2003 untill know and there is other way to find it too but i choose a easy one but is 2025 paper  which is explanation is so beeter than other paper, also some people write a package for take out this test with a second but maybe is not work for all i  search for that  but i didn't find it i will ask the question how we can find thus as shown in graph i did my train but need a little help while i am collect more information and style of solving 

Download p1.mw

How make simplify the fractional expresion?...

Asked by: salim-barzani 1635
radical simplify simplification
August 15 2025
2  1

In thus manuscript i got some reviewer comment which is asked to simplify this expresion and there is a lot of them maybe if i do by hand i  made a mistake becuase a lot of variable so how i can fix this issue and make thus square root are very simple as they demand

> 

restart
> 

B[2] := 0

0

 (1)

K := sqrt(-(1/2)*sqrt(2)*sqrt(lambda*a[5]/a[4])+sqrt(-a[5]/(2*a[4]))*(B[1]*sqrt(-lambda)*sinh(xi*sqrt(-lambda))+B[2]*sqrt(-lambda)*cosh(xi*sqrt(-lambda)))/(B[1]*cosh(xi*sqrt(-lambda))+B[2]*sinh(xi*sqrt(-lambda))+mu/lambda)+sqrt(-(lambda^2*B[1]^2*a[5]-lambda^2*B[2]^2*a[5]-mu^2*a[5])/(2*lambda*a[4]))/(B[1]*cosh(xi*sqrt(-lambda))+B[2]*sinh(xi*sqrt(-lambda))+mu/lambda))*exp(I*(k*(xi+v*tau^alpha/alpha)+w*tau^alpha/alpha+gamma))

(1/2)*(-2*2^(1/2)*(lambda*a[5]/a[4])^(1/2)+2*(-2*a[5]/a[4])^(1/2)*B[1]*(-lambda)^(1/2)*sinh(xi*(-lambda)^(1/2))/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda)+2*(-2*(lambda^2*B[1]^2*a[5]-mu^2*a[5])/(lambda*a[4]))^(1/2)/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda))^(1/2)*exp(I*(k*(xi+v*tau^alpha/alpha)+w*tau^alpha/alpha+gamma))

 (2)
> 

simplify(K)

(1/2)*exp(I*((k*v+w)*tau^alpha+alpha*(k*xi+gamma))/alpha)*2^(3/4)*((lambda*(a[5]*(-lambda^2*B[1]^2+mu^2)/(lambda*a[4]))^(1/2)+(-B[1]*cosh(xi*(-lambda)^(1/2))*lambda-mu)*(lambda*a[5]/a[4])^(1/2)+sinh(xi*(-lambda)^(1/2))*lambda*(-a[5]/a[4])^(1/2)*(-lambda)^(1/2)*B[1])/(B[1]*cosh(xi*(-lambda)^(1/2))*lambda+mu))^(1/2)

 (3)

subsindets(K, `&*`(rational, anything^(1/2)), proc (u) options operator, arrow; (u^2)^(1/2) end proc)

(1/2)*(-2*2^(1/2)*(lambda*a[5]/a[4])^(1/2)+2*(-2*a[5]/a[4])^(1/2)*B[1]*(-lambda)^(1/2)*sinh(xi*(-lambda)^(1/2))/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda)+2*(-2*(lambda^2*B[1]^2*a[5]-mu^2*a[5])/(lambda*a[4]))^(1/2)/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda))^(1/2)*exp(I*(k*(xi+v*tau^alpha/alpha)+w*tau^alpha/alpha+gamma))

 (4)

latex(%)

\frac{\sqrt{-2 \sqrt{2}\, \sqrt{\frac{\lambda  a_{5}}{a_{4}}}+\frac{2 \sqrt{-\frac{2 a_{5}}{a_{4}}}\, B_{1} \sqrt{-\lambda}\, \sinh \left(\xi  \sqrt{-\lambda}\right)}{B_{1} \cosh \left(\xi  \sqrt{-\lambda}\right)+\frac{\mu}{\lambda}}+\frac{2 \sqrt{-\frac{2 \left(\lambda^{2} B_{1}^{2} a_{5}-\mu^{2} a_{5}\right)}{\lambda  a_{4}}}}{B_{1} \cosh \left(\xi  \sqrt{-\lambda}\right)+\frac{\mu}{\lambda}}}\, {\mathrm e}^{\mathrm{I} \left(k \left(\xi +\frac{v \,\tau^{\alpha}}{\alpha}\right)+\frac{w \,\tau^{\alpha}}{\alpha}+\gamma \right)}}{2}

  

KK := sqrt(-(1/2)*sqrt(2)*sqrt(lambda*a[5]/a[4])+sqrt(-a[5]/(2*a[4]))*(B[1]*sqrt(-lambda)*sinh(xi*sqrt(-lambda))+B[2]*sqrt(-lambda)*cosh(xi*sqrt(-lambda)))/(B[1]*cosh(xi*sqrt(-lambda))+B[2]*sinh(xi*sqrt(-lambda))+mu/lambda)+sqrt(-(lambda^2*B[1]^2*a[5]-lambda^2*B[2]^2*a[5]-mu^2*a[5])/(2*lambda*a[4]))/(B[1]*cosh(xi*sqrt(-lambda))+B[2]*sinh(xi*sqrt(-lambda))+mu/lambda))*exp(I*(k*(xi+v*tau^alpha/alpha)+w*tau^alpha/alpha+gamma))

(1/2)*(-2*2^(1/2)*(lambda*a[5]/a[4])^(1/2)+2*(-2*a[5]/a[4])^(1/2)*B[1]*(-lambda)^(1/2)*sinh(xi*(-lambda)^(1/2))/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda)+2*(-2*(lambda^2*B[1]^2*a[5]-mu^2*a[5])/(lambda*a[4]))^(1/2)/(B[1]*cosh(xi*(-lambda)^(1/2))+mu/lambda))^(1/2)*exp((k*(xi+v*tau^alpha/alpha)+w*tau^alpha/alpha+gamma)*I)

 (5)

latex(KK)

\frac{\sqrt{-2 \sqrt{2}\, \sqrt{\frac{\lambda  a_{5}}{a_{4}}}+\frac{2 \sqrt{-\frac{2 a_{5}}{a_{4}}}\, B_{1} \sqrt{-\lambda}\, \sinh \left(\xi  \sqrt{-\lambda}\right)}{B_{1} \cosh \left(\xi  \sqrt{-\lambda}\right)+\frac{\mu}{\lambda}}+\frac{2 \sqrt{-\frac{2 \left(\lambda^{2} B_{1}^{2} a_{5}-\mu^{2} a_{5}\right)}{\lambda  a_{4}}}}{B_{1} \cosh \left(\xi  \sqrt{-\lambda}\right)+\frac{\mu}{\lambda}}}\, {\mathrm e}^{\mathrm{I} \left(k \left(\xi +\frac{v \,\tau^{\alpha}}{\alpha}\right)+\frac{w \,\tau^{\alpha}}{\alpha}+\gamma \right)}}{2}

  

NULL

Download simplify.mw

How separate linear and nonlinear of pdes which co...

Asked by: salim-barzani 1635
linear pde nonlinear
August 10 2025
1  15

i  can determine the pdes by one variable which is work so good but in some of the pdes i have two function i can separate by hand but how i can do by maple?

Download linear.mw

There is any way for avoid from `[Length of output...

Asked by: salim-barzani 1635
August 10 2025
0  7

i did a lot of trail to avoid for find my parameter in the last step of that i get this `[Length of output exceeds limit of 1000000]` and i don't know how to fix it i need to find that parameter but when i do substitution  is said this there is any way for hundle this situation 

help-parameter.mw

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