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Question: Collocation method

Question:Collocation method

Madhukesh J K 140 Maple 2017
ode homework numeric differential-equations bvp
August 02 2025
0

How to solve the given system of equations using the collocation method. I already tried numerical method. 

Kindly help me.

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restart:
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with(plots):
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M := 0.1e-1;

0.1e-1

  

6.3

  

1

 (1)
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de1 := diff(y(x), `$`(x, 3))+y(x)*(diff(y(x), `$`(x, 2)))-(diff(y(x), `$`(x, 2)))^2-M*(diff(y(x), x)) = 0;

diff(diff(diff(y(x), x), x), x)+y(x)*(diff(diff(y(x), x), x))-(diff(diff(y(x), x), x))^2-0.1e-1*(diff(y(x), x)) = 0

 (2)
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de2 := (diff(z(x), `$`(x, 2)))/Pr+y(x)*(diff(z(x), x))+Hs*z(x) = 0

.1587301587*(diff(diff(z(x), x), x))+y(x)*(diff(z(x), x))+z(x) = 0

 (3)
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bc := y(0) = 0, (D(y))(0) = 1, (D(y))(5) = 0, z(0) = 1, z(5) = 0;

y(0) = 0, (D(y))(0) = 1, (D(y))(5) = 0, z(0) = 1, z(5) = 0

 (4)
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sol := dsolve(eval([de1, de2, bc]), numeric);

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(50, {(1) = .0, (2) = 0.538450712436539e-1, (3) = .1082766859915131, (4) = .16331444031270473, (5) = .2189790463606035, (6) = .2752924234461127, (7) = .33227779889470094, (8) = .38995982000331736, (9) = .4483646786278111, (10) = .5075202501873479, (11) = .5674562491788733, (12) = .6282044036635384, (13) = .6897986516330131, (14) = .7524176453724036, (15) = .816715733668989, (16) = .8829036702135253, (17) = .9511585880378444, (18) = 1.0216871515791115, (19) = 1.0947329467114486, (20) = 1.170586433436612, (21) = 1.2495986390793312, (22) = 1.3322004686809672, (23) = 1.4189307352844525, (24) = 1.5100425310518173, (25) = 1.605556044446249, (26) = 1.706170701849191, (27) = 1.8128032039007311, (28) = 1.9266829029680352, (29) = 2.049525210932362, (30) = 2.1836360719075385, (31) = 2.3285917464945434, (32) = 2.485839068288575, (33) = 2.6591632259916818, (34) = 2.8548144648261875, (35) = 3.0721020019341, (36) = 3.305701178649673, (37) = 3.559949463816246, (38) = 3.7970789219278824, (39) = 3.990153302379728, (40) = 4.15720355507037, (41) = 4.306208343595448, (42) = 4.427174986030093, (43) = 4.526423733667591, (44) = 4.612636841962476, (45) = 4.689904237648306, (46) = 4.7605405732587975, (47) = 4.826006890793816, (48) = 4.88729546673539, (49) = 4.945116721690995, (50) = 5.0}, datatype = float[8], order = C_order); Y := Matrix(50, 5, {(1, 1) = .0, (1, 2) = 1.0, (1, 3) = -1.016912631477338, (1, 4) = 1.0, (1, 5) = 4.938383111325546, (2, 1) = 0.5239768838140454e-1, (2, 2) = .9467292622483128, (2, 3) = -.9622772113928727, (2, 4) = 1.2552137152775404, (2, 5) = 4.513346170739063, (3, 1) = .10253029316475297, (3, 2) = .895783706943777, (3, 3) = -.9101344320647631, (3, 4) = 1.4856388355002517, (3, 5) = 3.9305405877166715, (4, 1) = .1504792351703645, (4, 2) = .8470745536097423, (4, 3) = -.860373451742725, (4, 4) = 1.6829551945815935, (4, 5) = 3.223524805389885, (5, 1) = .19632323217740202, (5, 2) = .8005166268083528, (5, 3) = -.8128903023040563, (5, 4) = 1.8406334092867063, (5, 5) = 2.4329926016781056, (6, 1) = .24013840636663933, (6, 2) = .75602811722521, (6, 3) = -.7675871163117387, (6, 4) = 1.9543101701250922, (6, 5) = 1.6030729437227047, (7, 1) = .2819983835793791, (7, 2) = .7135303716905912, (7, 3) = -.7243714778269336, (7, 4) = 2.021973234344372, (7, 5) = .7775670166921098, (8, 1) = .3219743854445238, (8, 2) = .6729477074610013, (8, 3) = -.6831558741457402, (8, 4) = 2.04395171767529, (8, 5) = -0.34653255090257517e-2, (9, 1) = .36013531525388137, (9, 2) = .6342072469341395, (9, 3) = -.6438572302054374, (9, 4) = 2.0227281462789084, (9, 5) = -.7064433677651218, (10, 1) = .3965478383244555, (10, 2) = .5972387696518396, (10, 3) = -.6063965109971585, (10, 4) = 1.9626055145558932, (10, 5) = -1.3061198107940517, (11, 1) = .4312764574697176, (11, 2) = .5619745789945647, (11, 3) = -.5706983801552904, (11, 4) = 1.8692736251817776, (11, 5) = -1.786473233829117, (12, 1) = .46438358410487035, (12, 2) = .528349381415236, (12, 3) = -.5366909051394194, (12, 4) = 1.7493236289925442, (12, 5) = -2.1406810006817762, (13, 1) = .49592960543014786, (13, 2) = .49630017642071433, (13, 3) = -.5043053012120328, (13, 4) = 1.6097582709295248, (13, 5) = -2.3703112091517946, (14, 1) = .5260392137265275, (14, 2) = .46569879325333513, (14, 3) = -.4734077198356524, (14, 4) = 1.4571855527545918, (14, 5) = -2.4840706437151705, (15, 1) = .5550249899686894, (15, 2) = .4362259970506212, (15, 3) = -.44367242988215233, (15, 4) = 1.2966009526912445, (15, 5) = -2.4948251356156317, (16, 1) = .582947323077621, (16, 2) = .4078186772916242, (16, 3) = -.41503308219069956, (16, 4) = 1.133618168947713, (16, 5) = -2.4170112399060972, (17, 1) = .6098379764681325, (17, 2) = .3804427617884565, (17, 3) = -.3874529846540737, (17, 4) = .9734125833836331, (17, 5) = -2.267713799142949, (18, 1) = .6357288212995027, (18, 2) = .35406397565685854, (18, 3) = -.3608955340280772, (18, 4) = .8203928722713415, (18, 5) = -2.0653717123140347, (19, 1) = .6606520558243368, (19, 2) = .32864756458000216, (19, 3) = -.33532390992055233, (19, 4) = .6780764948146691, (19, 5) = -1.8284211343980543, (20, 1) = .6846405020881483, (20, 2) = .30415792526822016, (20, 3) = -.310700681058048, (20, 4) = .5490330762061609, (20, 5) = -1.5741295655423746, (21, 1) = .7077280179775027, (21, 2) = .2805580974006707, (21, 3) = -.2869872780487047, (21, 4) = .4348909698491685, (21, 5) = -1.3176869854655087, (22, 1) = .7299500860736533, (22, 2) = .25780904339388555, (22, 3) = -.2641432592769831, (22, 4) = .33639815254557953, (22, 5) = -1.071588245625372, (23, 1) = .7513446824807472, (23, 2) = .2358685927532746, (23, 3) = -.2421252475321414, (23, 4) = .2535257527268136, (23, 5) = -.845310470542935, (24, 1) = .7718600373416271, (24, 2) = .2147861043754655, (24, 3) = -.2209818389817886, (24, 4) = .18588236390672697, (24, 5) = -.6461214157898081, (25, 1) = .7913984364444584, (25, 2) = .1946582172929835, (25, 3) = -.20080896597698863, (25, 4) = .13250101521230842, (25, 5) = -.47835610933187955, (26, 1) = .8100007854235601, (26, 2) = .1754386620756086, (26, 3) = -.18155930417548316, (26, 4) = 0.9154569608003324e-1, (26, 5) = -.3421381651798363, (27, 1) = .8277118076501322, (27, 2) = .15707596816855374, (27, 3) = -.16318052643948905, (27, 4) = 0.6105466126293412e-1, (27, 5) = -.23553722921336775, (28, 1) = .8445805217874315, (28, 2) = .13951234873701424, (28, 3) = -.14561421333147123, (28, 4) = 0.39087681110023476e-1, (28, 5) = -.15525672367643156, (29, 1) = .8606634863301678, (29, 2) = .12267930610986412, (29, 3) = -.12879150739256892, (29, 4) = 0.23832991063636576e-1, (29, 5) = -0.9724838130607329e-1, (30, 1) = .8760079352865905, (30, 2) = .10651370141941992, (30, 3) = -.1126492397968819, (30, 4) = 0.13691444825679132e-1, (30, 5) = -0.5727653723318041e-1, (31, 1) = .8903192581096104, (31, 2) = 0.9131052167272624e-1, (31, 3) = -0.9748175032923309e-1, (31, 4) = 0.74131272943985405e-2, (31, 5) = -0.3172088613075896e-1, (32, 1) = .9035329750439559, (32, 2) = 0.7712276927556598e-1, (32, 3) = -0.8334133909913169e-1, (32, 4) = 0.3755698608845979e-2, (32, 5) = -0.1640104680740198e-1, (33, 1) = .9157173948547132, (33, 2) = 0.6385549523046109e-1, (33, 3) = -0.7013295494960084e-1, (33, 4) = 0.17491668487648983e-2, (33, 5) = -0.7780056793655609e-2, (34, 1) = .9269514730577374, (34, 2) = 0.51386320628059305e-1, (34, 3) = -0.57734884774429875e-1, (34, 4) = 0.7271289957321933e-3, (34, 5) = -0.32884474837452868e-2, (35, 1) = .9368471128429581, (35, 2) = 0.4010243173621042e-1, (35, 3) = -0.4653193161278045e-1, (35, 4) = 0.27013268380656147e-3, (35, 5) = -0.12396732969434932e-2, (36, 1) = .9450380185477333, (36, 2) = 0.30399509190318585e-1, (36, 3) = -0.36915005251636664e-1, (36, 4) = 0.9182614553853198e-4, (36, 5) = -0.42674003103921183e-3, (37, 1) = .9516679912394145, (37, 2) = 0.22101471191290013e-1, (37, 3) = -0.28706677187926227e-1, (37, 4) = 0.27971094145282712e-4, (37, 5) = -0.1315582904251911e-3, (38, 1) = .9561612871284315, (38, 2) = 0.16032468497041725e-1, (38, 3) = -0.22716201697191084e-1, (38, 4) = 0.9108284297335907e-5, (38, 5) = -0.4337758039619726e-4, (39, 1) = .958858965635045, (39, 2) = 0.12038500162795706e-1, (39, 3) = -0.18782005182342264e-1, (39, 4) = 0.3608857075468415e-5, (39, 5) = -0.17441599450102097e-4, (40, 1) = .9606217197165348, (40, 2) = 0.914514325386327e-2, (40, 3) = -0.1593723322791538e-1, (40, 4) = 0.15973572885621025e-5, (40, 5) = -0.7881196067433009e-5, (41, 1) = .9618157889669987, (41, 2) = 0.693594504796013e-2, (41, 3) = -0.13768928625667299e-1, (41, 4) = 0.7578240406846033e-6, (41, 5) = -0.3856388483481057e-5, (42, 1) = .9625579336916471, (42, 2) = 0.53653148882890234e-2, (42, 3) = -0.12229837199143657e-1, (42, 4) = 0.40466831691457055e-6, (42, 5) = -0.21455723871656463e-5, (43, 1) = .9630321047416508, (43, 2) = 0.4208610204678467e-2, (43, 3) = -0.11097927826763769e-1, (43, 4) = 0.23599512230457527e-6, (43, 5) = -0.13185086023573497e-5, (44, 1) = .9633548330674003, (44, 2) = 0.3291032013106795e-2, (44, 3) = -0.10201119562353245e-1, (44, 4) = 0.14351096807189318e-6, (44, 5) = -0.8585799100739333e-6, (45, 1) = .9635794225798572, (45, 2) = 0.25318181431538732e-2, (45, 3) = -0.9459930001639555e-2, (45, 4) = 0.8857046151520711e-7, (45, 5) = -0.5807183930298681e-6, (46, 1) = .9637351936467563, (46, 2) = 0.18861028334058306e-2, (46, 3) = -0.8830217058260243e-2, (46, 4) = 0.54176953171370387e-7, (46, 5) = -0.40321204700634103e-6, (47, 1) = .9638401434562422, (47, 2) = 0.1326072836040432e-2, (47, 3) = -0.828462376949864e-2, (47, 4) = 0.318508859340672e-7, (47, 5) = -0.28512654551555526e-6, (48, 1) = .9639061617395365, (48, 2) = 0.8331693356556849e-3, (48, 3) = -0.7804900784463723e-2, (48, 4) = 0.16983012204335007e-7, (48, 5) = -0.20411108581207533e-6, (49, 1) = .963941530746681, (49, 2) = 0.39433359612162176e-3, (49, 3) = -0.7378210574196815e-2, (49, 4) = 0.69058956940428584e-8, (49, 5) = -0.14716673870893628e-6, (50, 1) = .963952255758426, (50, 2) = .0, (50, 3) = -0.6995152048687603e-2, (50, 4) = .0, (50, 5) = -0.10635754806174274e-6}, datatype = float[8], order = C_order); YP := Matrix(50, 5, {(1, 1) = 1.0, (1, 2) = -1.016912631477338, (1, 3) = 1.0441113000581643, (1, 4) = 4.938383111325546, (1, 5) = -6.300000001197, (2, 1) = .9467292622483128, (2, 2) = -.9622772113928727, (2, 3) = .9858658256476173, (2, 4) = 4.513346170739063, (2, 5) = -9.397726517168355, (3, 1) = .895783706943777, (3, 2) = -.9101344320647631, (3, 3) = .9306188716382228, (3, 4) = 3.9305405877166715, (3, 5) = -11.898421382065955, (4, 1) = .8470745536097423, (4, 2) = -.860373451742725, (4, 3) = .8781815609789202, (4, 4) = 3.223524805389885, (4, 5) = -13.658581076246108, (5, 1) = .8005166268083528, (5, 2) = -.8128903023040563, (5, 3) = .8283850614020614, (5, 4) = 2.4329926016781056, (5, 5) = -14.605204201259696, (6, 1) = .75602811722521, (6, 2) = -.7675871163117387, (6, 3) = .7810774091586878, (6, 4) = 1.6030729437227047, (6, 5) = -14.737398181156987, (7, 1) = .7135303716905912, (7, 2) = -.7243714778269336, (7, 3) = .736120927464283, (7, 4) = .7775670166921098, (7, 5) = -14.11984902259275, (8, 1) = .6729477074610013, (8, 2) = -.6831558741457402, (8, 3) = .6933901181953316, (8, 4) = -0.34653255090257517e-2, (8, 5) = -12.86986662367746, (9, 1) = .6342072469341395, (9, 2) = -.6438572302054374, (9, 3) = .6527699319356851, (9, 4) = -.7064433677651218, (9, 5) = -11.140371532431427, (10, 1) = .5972387696518396, (10, 2) = -.6063965109971585, (10, 3) = .6141543418494605, (10, 4) = -1.3061198107940517, (10, 5) = -9.101399121783686, (11, 1) = .5619745789945647, (11, 2) = -.5706983801552904, (11, 3) = .5774451625788978, (11, 4) = -1.786473233829117, (11, 5) = -6.922501599763635, (12, 1) = .528349381415236, (12, 2) = -.5366909051394194, (12, 3) = .5425510675586522, (12, 4) = -2.1406810006817762, (12, 5) = -4.757927035769672, (13, 1) = .49630017642071433, (13, 2) = -.5043053012120328, (13, 3) = .5093867676411815, (13, 4) = -2.3703112091517946, (13, 5) = -2.7357798403575755, (14, 1) = .46569879325333513, (14, 2) = -.4734077198356524, (14, 3) = .47780288184693964, (14, 4) = -2.4840706437151705, (14, 5) = -.9479420024892491, (15, 1) = .4362259970506212, (15, 2) = -.44367242988215233, (15, 3) = .44745677095276526, (15, 4) = -2.4948251356156317, (15, 5) = .5549628621234142, (16, 1) = .4078186772916242, (16, 2) = -.41503308219069956, (16, 3) = .4182730703373508, (16, 4) = -2.4170112399060972, (16, 5) = 1.7348439985152488, (17, 1) = .3804427617884565, (17, 2) = -.3874529846540737, (17, 3) = .39020778707321313, (17, 4) = -2.267713799142949, (17, 5) = 2.5800100903859544, (18, 1) = .35406397565685854, (18, 2) = -.3608955340280772, (18, 3) = .36321791869790376, (18, 4) = -2.0653717123140347, (18, 5) = 3.10352774783306, (19, 1) = .32864756458000216, (19, 2) = -.33532390992055233, (19, 3) = .3372610306662744, (19, 4) = -1.8284211343980543, (19, 5) = 3.338204225824109, (20, 1) = .30415792526822016, (20, 2) = -.310700681058048, (20, 3) = .31229476274132867, (20, 4) = -1.5741295655423746, (20, 5) = 3.3306826139938144, (21, 1) = .2805580974006707, (21, 2) = -.2869872780487047, (21, 3) = .28827621621397953, (21, 4) = -1.3176869854655087, (21, 5) = 3.135340081336959, (22, 1) = .25780904339388555, (22, 2) = -.2641432592769831, (22, 3) = .2651611467004155, (22, 4) = -1.071588245625372, (22, 5) = 2.8085890119139396, (23, 1) = .2358685927532746, (23, 2) = -.2421252475321414, (23, 3) = .2429028386476426, (23, 4) = -.845310470542935, (23, 5) = 2.4040407789305567, (24, 1) = .2147861043754655, (24, 2) = -.2209818389817886, (24, 3) = .2215478846918326, (24, 4) = -.6461214157898081, (24, 5) = 1.9708474985101825, (25, 1) = .1946582172929835, (25, 2) = -.20080896597698863, (25, 3) = .20119072468789445, (25, 4) = -.47835610933187955, (25, 5) = 1.5502363494870965, (26, 1) = .1754386620756086, (26, 2) = -.18155930417548316, (26, 3) = .18178134653653813, (26, 4) = -.3421381651798363, (26, 5) = 1.1691948647879093, (27, 1) = .15707596816855374, (27, 2) = -.16318052643948905, (27, 3) = .163265092403284, (27, 4) = -.23553722921336775, (27, 5) = .843584392498727, (28, 1) = .13951234873701424, (28, 2) = -.14561421333147123, (28, 3) = .14558155088667368, (28, 4) = -.15525672367643156, (28, 5) = .579846478687003, (29, 1) = .12267930610986412, (29, 2) = -.12879150739256892, (29, 3) = .12866019319975472, (29, 4) = -0.9724838130607329e-1, (29, 5) = .3771503810083066, (30, 1) = .10651370141941992, (30, 2) = -.1126492397968819, (30, 3) = .11243661620708013, (30, 4) = -0.5727653723318041e-1, (30, 5) = .22984451471051648, (31, 1) = 0.9131052167272624e-1, (31, 2) = -0.9748175032923309e-1, (31, 3) = 0.9720567649632728e-1, (31, 4) = -0.3172088613075896e-1, (31, 5) = .13122010765127654, (32, 1) = 0.7712276927556598e-1, (32, 2) = -0.8334133909913169e-1, (32, 3) = 0.8301865455597775e-1, (32, 4) = -0.1640104680740198e-1, (32, 5) = 0.6969808445659362e-1, (33, 1) = 0.6385549523046109e-1, (33, 2) = -0.7013295494960084e-1, (33, 3) = 0.697791531221788e-1, (33, 4) = -0.7780056793655609e-2, (33, 5) = 0.3386354889433578e-1, (34, 1) = 0.51386320628059305e-1, (34, 2) = -0.57734884774429875e-1, (34, 3) = 0.5736461661467379e-1, (34, 4) = -0.32884474837452868e-2, (34, 5) = 0.14622944136188972e-1, (35, 1) = 0.4010243173621042e-1, (35, 2) = -0.4653193161278045e-1, (35, 3) = 0.4615955076341791e-1, (35, 4) = -0.12396732969434932e-2, (35, 5) = 0.5614885492478633e-2, (36, 1) = 0.30399509190318585e-1, (36, 2) = -0.36915005251636664e-1, (36, 3) = 0.3655279612231743e-1, (36, 4) = -0.42674003103921183e-3, (36, 5) = 0.19621942697003233e-2, (37, 1) = 0.22101471191290013e-1, (37, 2) = -0.28706677187926227e-1, (37, 3) = 0.2836431384167678e-1, (37, 4) = -0.1315582904251911e-3, (37, 5) = 0.6125409350740502e-3, (38, 1) = 0.16032468497041725e-1, (38, 2) = -0.22716201697191084e-1, (38, 3) = 0.22396703157973173e-1, (38, 4) = -0.4337758039619726e-4, (38, 5) = 0.20391637652164134e-3, (39, 1) = 0.12038500162795706e-1, (39, 2) = -0.18782005182342264e-1, (39, 3) = 0.18482442781990245e-1, (39, 4) = -0.17441599450102097e-4, (39, 5) = 0.8262561468904556e-4, (40, 1) = 0.914514325386327e-2, (40, 2) = -0.1593723322791538e-1, (40, 3) = 0.15655099226423174e-1, (40, 4) = -0.7881196067433009e-5, (40, 5) = 0.3763299224344935e-4, (41, 1) = 0.693594504796013e-2, (41, 2) = -0.13768928625667299e-1, (41, 3) = 0.13502115795304808e-1, (41, 4) = -0.3856388483481057e-5, (41, 5) = 0.18593261137575973e-4, (42, 1) = 0.53653148882890234e-2, (42, 2) = -0.12229837199143657e-1, (42, 3) = 0.11975148890593406e-1, (42, 4) = -0.21455723871656463e-5, (42, 5) = 0.10461587263954827e-4, (43, 1) = 0.4208610204678467e-2, (43, 2) = -0.11097927826763769e-1, (43, 3) = 0.1085291089737409e-1, (43, 4) = -0.13185086023573497e-5, (43, 5) = 0.6512757251742076e-5, (44, 1) = 0.3291032013106795e-2, (44, 2) = -0.10201119562353245e-1, (44, 3) = 0.9964270993547896e-2, (44, 4) = -0.8585799100739333e-6, (44, 5) = 0.4306718669414423e-5, (45, 1) = 0.25318181431538732e-2, (45, 2) = -0.9459930001639555e-2, (45, 3) = 0.9230202345693167e-2, (45, 4) = -0.5807183930298681e-6, (45, 5) = 0.29672863441924835e-5, (46, 1) = 0.18861028334058306e-2, (46, 2) = -0.8830217058260243e-2, (46, 3) = 0.8606824708215374e-2, (46, 4) = -0.40321204700634103e-6, (46, 5) = 0.2106799928695533e-5, (47, 1) = 0.1326072836040432e-2, (47, 2) = -0.828462376949864e-2, (47, 3) = 0.8066948681837108e-2, (47, 4) = -0.28512654551555526e-6, (47, 5) = 0.15306828052634482e-5, (48, 1) = 0.8331693356556849e-3, (48, 2) = -0.7804900784463723e-2, (48, 3) = 0.75924401275222035e-2, (48, 4) = -0.20411108581207533e-6, (48, 5) = 0.11324938030775855e-5, (49, 1) = 0.39433359612162176e-3, (49, 2) = -0.7378210574196815e-2, (49, 3) = 0.7170544922301032e-2, (49, 4) = -0.14716673870893628e-6, (49, 5) = 0.85021168502143e-6, (50, 1) = .0, (50, 2) = -0.6995152048687603e-2, (50, 3) = 0.6791924748889848e-2, (50, 4) = -0.10635754806174274e-6, (50, 5) = 0.6458986698603491e-6}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(50, {(1) = .0, (2) = 0.538450712436539e-1, (3) = .1082766859915131, (4) = .16331444031270473, (5) = .2189790463606035, (6) = .2752924234461127, (7) = .33227779889470094, (8) = .38995982000331736, (9) = .4483646786278111, (10) = .5075202501873479, (11) = .5674562491788733, (12) = .6282044036635384, (13) = .6897986516330131, (14) = .7524176453724036, (15) = .816715733668989, (16) = .8829036702135253, (17) = .9511585880378444, (18) = 1.0216871515791115, (19) = 1.0947329467114486, (20) = 1.170586433436612, (21) = 1.2495986390793312, (22) = 1.3322004686809672, (23) = 1.4189307352844525, (24) = 1.5100425310518173, (25) = 1.605556044446249, (26) = 1.706170701849191, (27) = 1.8128032039007311, (28) = 1.9266829029680352, (29) = 2.049525210932362, (30) = 2.1836360719075385, (31) = 2.3285917464945434, (32) = 2.485839068288575, (33) = 2.6591632259916818, (34) = 2.8548144648261875, (35) = 3.0721020019341, (36) = 3.305701178649673, (37) = 3.559949463816246, (38) = 3.7970789219278824, (39) = 3.990153302379728, (40) = 4.15720355507037, (41) = 4.306208343595448, (42) = 4.427174986030093, (43) = 4.526423733667591, (44) = 4.612636841962476, (45) = 4.689904237648306, (46) = 4.7605405732587975, (47) = 4.826006890793816, (48) = 4.88729546673539, (49) = 4.945116721690995, (50) = 5.0}, datatype = float[8], order = C_order); Y := Matrix(50, 5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = 0.15143863606013115e-13, (1, 4) = .0, (1, 5) = 0.5209145281927147e-8, (2, 1) = -0.8264612893080438e-16, (2, 2) = 0.10601022236424583e-14, (2, 3) = 0.13448281300622448e-13, (2, 4) = 0.2531723244828144e-9, (2, 5) = 0.5151764399789821e-8, (3, 1) = -0.9818312986936232e-16, (3, 2) = 0.2024546298845675e-14, (3, 3) = 0.11603414280085763e-13, (3, 4) = 0.5031439996715309e-9, (3, 5) = 0.49505227176451794e-8, (4, 1) = -0.16400920960349306e-15, (4, 2) = 0.22923225644852182e-14, (4, 3) = 0.9858996413953448e-14, (4, 4) = 0.7435382816273587e-9, (4, 5) = 0.4613790921382858e-8, (5, 1) = 0.16695611005256184e-15, (5, 2) = 0.2920750522546946e-14, (5, 3) = 0.8715810892933721e-14, (5, 4) = 0.968257325936286e-9, (5, 5) = 0.4156299767076735e-8, (6, 1) = 0.3785488577501235e-15, (6, 2) = 0.34710943442817573e-14, (6, 3) = 0.7739663648523766e-14, (6, 4) = 0.11717218646887862e-8, (6, 5) = 0.359812314472648e-8, (7, 1) = 0.5735291564092602e-15, (7, 2) = 0.5559621333708256e-14, (7, 3) = 0.64794162714847054e-14, (7, 4) = 0.1349065434755046e-8, (7, 5) = 0.29635492365957813e-8, (8, 1) = 0.7392908645452232e-15, (8, 2) = 0.3999400361865862e-14, (8, 3) = 0.6347742731734012e-14, (8, 4) = 0.14963058193845247e-8, (8, 5) = 0.2279713504762398e-8, (9, 1) = 0.6831780723045333e-15, (9, 2) = 0.5442883341070182e-14, (9, 3) = 0.56555333890748366e-14, (9, 4) = 0.1610546333661808e-8, (9, 5) = 0.15751280801152935e-8, (10, 1) = 0.12805951228799598e-14, (10, 2) = 0.5552725838454675e-14, (10, 3) = 0.494213125294359e-14, (10, 4) = 0.16900532451603174e-8, (10, 5) = 0.8780983197662242e-9, (11, 1) = 0.1589045835010457e-14, (11, 2) = 0.54573811395900386e-14, (11, 3) = 0.4370346233896689e-14, (11, 4) = 0.1734384141772804e-8, (11, 5) = 0.21511245732875356e-9, (12, 1) = 0.1938397406295573e-14, (12, 2) = 0.65363028178274735e-14, (12, 3) = 0.39915199224484734e-14, (12, 4) = 0.17443804487181745e-8, (12, 5) = -0.3906337574991102e-9, (13, 1) = 0.25746355641282846e-14, (13, 2) = 0.6686301295711723e-14, (13, 3) = 0.3776993739734716e-14, (13, 4) = 0.17221022298538633e-8, (13, 5) = -0.9204760276081012e-9, (14, 1) = 0.24021854692179305e-14, (14, 2) = 0.68388696506603025e-14, (14, 3) = 0.3589227807197002e-14, (14, 4) = 0.16706007423685053e-8, (14, 5) = -0.13619565204084925e-8, (15, 1) = 0.27844846589144917e-14, (15, 2) = 0.6806482863864933e-14, (15, 3) = 0.3255802330557269e-14, (15, 4) = 0.15931710762978012e-8, (15, 5) = -0.1710057839046858e-8, (16, 1) = 0.3170168392339236e-14, (16, 2) = 0.7058625852330826e-14, (16, 3) = 0.3059882244808441e-14, (16, 4) = 0.14936064638531152e-8, (16, 5) = -0.19605804146242323e-8, (17, 1) = 0.418450905753915e-14, (17, 2) = 0.7320624596592493e-14, (17, 3) = 0.25290623286820664e-14, (17, 4) = 0.1376224007726725e-8, (17, 5) = -0.2113726736115561e-8, (18, 1) = 0.522802446903785e-14, (18, 2) = 0.7583088387046541e-14, (18, 3) = 0.2656436255857598e-14, (18, 4) = 0.12456454647509306e-8, (18, 5) = -0.2174149264673966e-8, (19, 1) = 0.5599963613653709e-14, (19, 2) = 0.7792902420006349e-14, (19, 3) = 0.2298977816983502e-14, (19, 4) = 0.1106554758169108e-8, (19, 5) = -0.21502682643665226e-8, (20, 1) = 0.5865892702136685e-14, (20, 2) = 0.8092855141368515e-14, (20, 3) = 0.2509193086994891e-14, (20, 4) = 0.9634933725685698e-9, (20, 5) = -0.205341703724594e-8, (21, 1) = 0.6816420131570755e-14, (21, 2) = 0.7846294298633722e-14, (21, 3) = 0.22775440990284527e-14, (21, 4) = 0.8206611972083126e-9, (21, 5) = -0.1896857216112942e-8, (22, 1) = 0.7301012092886368e-14, (22, 2) = 0.7949612920729695e-14, (22, 3) = 0.19113297564801313e-14, (22, 4) = 0.6817767801598973e-9, (22, 5) = -0.16947933473596584e-8, (23, 1) = 0.7517850272435388e-14, (23, 2) = 0.844440889491318e-14, (23, 3) = 0.210172135546369e-14, (23, 4) = 0.5499764453689152e-9, (23, 5) = -0.14614844738189918e-8, (24, 1) = 0.9093082110941868e-14, (24, 2) = 0.8617808603338214e-14, (24, 3) = 0.1973966157355317e-14, (24, 4) = 0.4281530842561462e-9, (24, 5) = -0.12106705564003452e-8, (25, 1) = 0.9614190502196018e-14, (25, 2) = 0.8848373195710656e-14, (25, 3) = 0.2086102522252812e-14, (25, 4) = 0.3188261735784484e-9, (25, 5) = -0.9556847653193206e-9, (26, 1) = 0.1041568490829814e-13, (26, 2) = 0.893833612887857e-14, (26, 3) = 0.1856000845238865e-14, (26, 4) = 0.22345041334248272e-9, (26, 5) = -0.708864527271831e-9, (27, 1) = 0.1152901561368169e-13, (27, 2) = 0.89795369925252e-14, (27, 3) = 0.2071787429729606e-14, (27, 4) = 0.14287136200501294e-9, (27, 5) = -0.4806758343384607e-9, (28, 1) = 0.13495965110034514e-13, (28, 2) = 0.9150624849596765e-14, (28, 3) = 0.18643959445317172e-14, (28, 4) = 0.7730346102119082e-10, (28, 5) = -0.27925524569939634e-9, (29, 1) = 0.1447846668412635e-13, (29, 2) = 0.9061416074595552e-14, (29, 3) = 0.2277139811068211e-14, (29, 4) = 0.26236748156626518e-10, (29, 5) = -0.10967840766932284e-9, (30, 1) = 0.1637043045937391e-13, (30, 2) = 0.8735733718415329e-14, (30, 3) = 0.27774780827091533e-14, (30, 4) = -0.11566251997979936e-10, (30, 5) = 0.26584437011128173e-10, (31, 1) = 0.19157667101419045e-13, (31, 2) = 0.8123059896172982e-14, (31, 3) = 0.36940270461759766e-14, (31, 4) = -0.3338327910770145e-10, (31, 5) = 0.11387033083646781e-9, (32, 1) = 0.218389656225817e-13, (32, 2) = 0.6857924190366744e-14, (32, 3) = 0.5214825473490382e-14, (32, 4) = -0.40398750655901706e-10, (32, 5) = 0.14926462934637388e-9, (33, 1) = 0.2627628563699054e-13, (33, 2) = 0.401556619991872e-14, (33, 3) = 0.8152745438393999e-14, (33, 4) = -0.3804768632584147e-10, (33, 5) = 0.14874007073963709e-9, (34, 1) = 0.36370999494014086e-13, (34, 2) = -0.386548921701265e-14, (34, 3) = 0.1631808180318328e-13, (34, 4) = -0.33000222761584506e-10, (34, 5) = 0.13693032069292921e-9, (35, 1) = 0.5509262327612127e-13, (35, 2) = -0.18890089391438668e-13, (35, 3) = 0.3161707194631968e-13, (35, 4) = -0.16646067039944626e-10, (35, 5) = 0.7219686567800556e-10, (36, 1) = 0.7409383620236018e-13, (36, 2) = -0.36040651044841214e-13, (36, 3) = 0.4898335208487011e-13, (36, 4) = 0.32784502413364445e-11, (36, 5) = -0.14735420939556755e-10, (37, 1) = 0.10169362342003957e-12, (37, 2) = -0.6040877776984526e-13, (37, 3) = 0.7350859063809685e-13, (37, 4) = 0.6987192987396587e-11, (37, 5) = -0.31593308676975374e-10, (38, 1) = 0.9107063627312493e-13, (38, 2) = -0.46451392159979334e-13, (38, 3) = 0.5970703758877559e-13, (38, 4) = 0.31960324368762172e-11, (38, 5) = -0.14549863332219355e-10, (39, 1) = 0.7795348303548032e-13, (39, 2) = -0.307504836025312e-13, (39, 3) = 0.4400477767345146e-13, (39, 4) = -0.24540000756116013e-12, (39, 5) = 0.14204634414083795e-11, (40, 1) = 0.7037299697401581e-13, (40, 2) = -0.2113986584686098e-13, (40, 3) = 0.3446868361893824e-13, (40, 4) = -0.6711604095422506e-12, (40, 5) = 0.33730077034860774e-11, (41, 1) = 0.6567420110784044e-13, (41, 2) = -0.1487071052735246e-13, (41, 3) = 0.281975245609803e-13, (41, 4) = -0.4840307488210744e-12, (41, 5) = 0.24565250659311624e-11, (42, 1) = 0.6459712022715409e-13, (42, 2) = -0.11097162630862281e-13, (42, 3) = 0.24423024403583574e-13, (42, 4) = -0.28608148225793283e-12, (42, 5) = 0.14964581624333766e-11, (43, 1) = 0.640280502136775e-13, (43, 2) = -0.8548606846495986e-14, (43, 3) = 0.21881466472077222e-13, (43, 4) = -0.16940774104908874e-12, (43, 5) = 0.9262688645940507e-12, (44, 1) = 0.6244301847516393e-13, (44, 2) = -0.66219479488107194e-14, (44, 3) = 0.1995701245760364e-13, (44, 4) = -0.10237686507119983e-12, (44, 5) = 0.5948441235731513e-12, (45, 1) = 0.6185753483413617e-13, (45, 2) = -0.5051468771290501e-14, (45, 3) = 0.18395260491368303e-13, (45, 4) = -0.6239441019233403e-13, (45, 5) = 0.3942092903383439e-12, (46, 1) = 0.6165132946892792e-13, (46, 2) = -0.37441003592832135e-14, (46, 3) = 0.1709672398590452e-13, (46, 4) = -0.3763161671814335e-13, (46, 5) = 0.26764102662748244e-12, (47, 1) = 0.6198151724876583e-13, (47, 2) = -0.26188562204839776e-14, (47, 3) = 0.15967252338380992e-13, (47, 4) = -0.21819149987483318e-13, (47, 5) = 0.18495665706484231e-12, (48, 1) = 0.6094944363926368e-13, (48, 2) = -0.1640899643671393e-14, (48, 3) = 0.15000579561151843e-13, (48, 4) = -0.11483823318682457e-13, (48, 5) = 0.1293625847432827e-12, (49, 1) = 0.6143323589996598e-13, (49, 2) = -0.7734412660573863e-15, (49, 3) = 0.14134062292410381e-13, (49, 4) = -0.4614422689930712e-14, (49, 5) = 0.9108760894060782e-13, (50, 1) = 0.6135526225959751e-13, (50, 2) = .0, (50, 3) = 0.13373226403332908e-13, (50, 4) = .0, (50, 5) = 0.6421709636688081e-13}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[50] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(5.209145281927147e-9) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [5, 50, [y(x), diff(y(x), x), diff(diff(y(x), x), x), z(x), diff(z(x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[50] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[50] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(5, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(50, 5, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(5, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(50, 5, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[y(x), diff(y(x), x), diff(diff(y(x), x), x), z(x), diff(z(x), x)]'[i] = yout[i], i = 1 .. 5)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[50] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(5.209145281927147e-9) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [5, 50, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[50] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[50] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(50, 5, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(5, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0.}); `dsolve/numeric/hermite`(50, 5, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 5)] end proc, (2) = Array(0..0, {}), (3) = [x, y(x), diff(y(x), x), diff(diff(y(x), x), x), z(x), diff(z(x), x)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[y(x), diff(y(x), x), diff(diff(y(x), x), x), z(x), diff(z(x), x)]'[i] = res[i+1], i = 1 .. 5)] catch: error end try end proc

 (5)
> 

print(sol(0));

[x = 0., y(x) = HFloat(0.0), diff(y(x), x) = HFloat(0.9999999999999998), diff(diff(y(x), x), x) = HFloat(-1.0169126314773378), z(x) = HFloat(0.9999999999999998), diff(z(x), x) = HFloat(4.938383111325545)]

 (6)
> 

odeplot(sol, [x, diff(y(x), x)], x = 0 .. 5)

 
> 

odeplot(sol, [x, z(x)], x = 0 .. 5)

 
 

 

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