tomleslie

13776 Reputation

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14 years, 57 days

MaplePrimes Activity


These are answers submitted by tomleslie

can only be done numerically, as in the attached

  restart;
  with(plots):
  Pr:=1:
  etamax := 10;
  eq:= 3/4*F(eta)*diff(theta(eta),eta) = diff(theta(eta),eta,eta),
       1/Pr*(1/2*diff(F(eta),eta)^2 - 3/4*F(eta)*diff(F(eta),eta,eta)) = -diff(F(eta),eta,eta,eta) + theta(eta);
  bcs := F(0) = 0, theta(0)=1, D(F)(0)=0,theta(etamax)=0,D(F)(etamax)=0;
  sol:=dsolve( [ eq, bcs ],
               [ F(eta), theta(eta)],
               numeric
             );
  odeplot( sol,
           [ [ eta, F(eta)],
             [eta, theta(eta) ]
           ],
           eta=0..etamax,
           color=[ red, blue]
         );

10

 

(3/4)*F(eta)*(diff(theta(eta), eta)) = diff(diff(theta(eta), eta), eta), (1/2)*(diff(F(eta), eta))^2-(3/4)*F(eta)*(diff(diff(F(eta), eta), eta)) = -(diff(diff(diff(F(eta), eta), eta), eta))+theta(eta)

 

F(0) = 0, theta(0) = 1, (D(F))(0) = 0, theta(10) = 0, (D(F))(10) = 0

 

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(31, {(1) = .0, (2) = .3243000540262205, (3) = .6494459686383482, (4) = .9754443972533966, (5) = 1.3023020809731753, (6) = 1.6301555741550084, (7) = 1.9595249872617806, (8) = 2.2904837219758925, (9) = 2.623055009854351, (10) = 2.957225069568361, (11) = 3.292416584893846, (12) = 3.6284306063278815, (13) = 3.965273218644352, (14) = 4.302949344760717, (15) = 4.640942032851813, (16) = 4.978823623378006, (17) = 5.316594225819793, (18) = 5.654253949478462, (19) = 5.991736378579274, (20) = 6.328914836969517, (21) = 6.6657870456550175, (22) = 7.002353837596869, (23) = 7.338652555245001, (24) = 7.6748437956823565, (25) = 8.0109464436098, (26) = 8.346960569027788, (27) = 8.682730041734624, (28) = 9.016229528103496, (29) = 9.346894111650576, (30) = 9.674794880368847, (31) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(31, 5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = -.9315092250220919, (1, 4) = 1.0, (1, 5) = -.34648123848818996, (2, 1) = -0.4346529224884781e-1, (2, 2) = -.25157174296412993, (2, 3) = -.6266429386440938, (2, 4) = .8877399522080314, (2, 5) = -.3452245048768078, (3, 1) = -.15356307629470006, (3, 2) = -.41143505643517725, (3, 3) = -.3644299174643001, (3, 4) = .776521308615952, (3, 5) = -.3373854265507104, (4, 1) = -.30303612198023977, (4, 2) = -.4939285418533152, (4, 3) = -.14996847236693056, (4, 4) = .6691913669270281, (4, 5) = -.31924318469866025, (5, 1) = -.4693302221186412, (5, 2) = -.5146039711733079, (5, 3) = 0.15286461167066837e-1, (5, 4) = .5692851662987374, (5, 5) = -.29044644794066193, (6, 1) = -.6348959243606519, (6, 2) = -.4889090046306086, (6, 3) = .134080691208861, (6, 4) = .479946768467719, (6, 5) = -.25353092284001993, (7, 1) = -.7870666102366355, (7, 2) = -.430798678299882, (7, 3) = .21263948408802658, (7, 4) = .4031376227624315, (7, 5) = -.212609847242516, (8, 1) = -.917022026251486, (8, 2) = -.35198208297484934, (8, 3) = .2589494824064258, (8, 4) = .33956117377166395, (8, 5) = -.17198796501957242, (9, 1) = -1.0192538910763331, (9, 2) = -.26157835898450443, (9, 3) = .2813845520480499, (9, 4) = .28865227152310796, (9, 5) = -.1350057879520591, (10, 1) = -1.0907797173777394, (10, 2) = -.16617020973083912, (10, 3) = .287445472379921, (10, 4) = .24896253452761852, (10, 5) = -.10356868593432067, (11, 1) = -1.1303744029345197, (11, 2) = -0.70328056399142e-1, (11, 3) = .28309382614500034, (11, 4) = .21865664992987685, (11, 5) = -0.7828607488757705e-1, (12, 1) = -1.13819996465162, (12, 2) = 0.23159682629474765e-1, (12, 3) = .2726118845713157, (12, 4) = .19578047120402123, (12, 5) = -0.5878328825796292e-1, (13, 1) = -1.1151836128273132, (13, 2) = .11272089234083588, (13, 3) = .2587658127130804, (13, 4) = .17856190003396138, (13, 5) = -0.4419377987848922e-1, (14, 1) = -1.0626582050808495, (14, 2) = .1974998734523982, (14, 3) = .24316815314012719, (14, 4) = .16553780161960197, (14, 5) = -0.33522031258804846e-1, (15, 1) = -.9823260978346049, (15, 2) = .2769182605675772, (15, 3) = .226662356776616, (15, 4) = .1555765047127104, (15, 5) = -0.25853335713280336e-1, (16, 1) = -.8761446635822387, (16, 2) = .35063241699188924, (16, 3) = .2095785407979796, (16, 4) = .14781280520038523, (16, 5) = -0.20418369785624606e-1, (17, 1) = -.7460888795492298, (17, 2) = .4184566810307239, (17, 3) = .1919106490845611, (17, 4) = .1415970290390463, (17, 5) = -0.16617772409198768e-1, (18, 1) = -.594198913292964, (18, 2) = .48016806829400327, (18, 3) = .173457068311107, (18, 4) = .13645378150703344, (18, 5) = -0.14017723407396463e-1, (19, 1) = -.4226380130778863, (19, 2) = .5354415592106987, (19, 3) = .1538838140196576, (19, 4) = .13203166501997782, (19, 5) = -0.12320225018360472e-1, (20, 1) = -.2337431183279873, (20, 2) = .5838153588501531, (20, 3) = .1327386247119349, (20, 4) = .12806176574444494, (20, 5) = -0.11335102191701743e-1, (21, 1) = -0.29969322870601396e-1, (21, 2) = .6246747020405069, (21, 3) = .1094150967785294, (21, 4) = .1243228015692788, (21, 5) = -0.10960526694470533e-1, (22, 1) = .18599206187673545, (22, 2) = .6571695251212902, (22, 3) = 0.8309377267737389e-1, (22, 4) = .12061460158722655, (22, 5) = -0.11175927429701141e-1, (23, 1) = .4111445467958876, (23, 2) = .6801269015815865, (23, 3) = 0.5261941106453635e-1, (23, 4) = .11672975808513042, (23, 5) = -0.12048085031021012e-1, (24, 1) = .6421187842759781, (24, 2) = .691903987919029, (24, 3) = 0.16281531791945737e-1, (24, 4) = .11241921048906331, (24, 5) = -0.13757841466591127e-1, (25, 1) = .8747929250117037, (25, 2) = .6901321359389782, (25, 3) = -0.2852263328788306e-1, (25, 4) = .10734835326070424, (25, 5) = -0.16656662943528553e-1, (26, 1) = 1.10406798468371, (26, 2) = .6713342300948061, (26, 3) = -0.859299780498995e-1, (26, 4) = .10102195887950144, (26, 5) = -0.21376438602711868e-1, (27, 1) = 1.3233061270901374, (27, 2) = .6302642389051082, (27, 3) = -.16271243923470902, (27, 4) = 0.9266389880060642e-1, (27, 5) = -0.29026728270966817e-1, (28, 1) = 1.5226438194159508, (28, 2) = .559241686161435, (28, 3) = -.2696145103174367, (28, 4) = 0.810819065288907e-1, (28, 5) = -0.4145590669149437e-1, (29, 1) = 1.6902721278742945, (29, 2) = .4461023878388593, (29, 3) = -.42517586122009443, (29, 4) = 0.6429263539520742e-1, (29, 5) = -0.6179362826521952e-1, (30, 1) = 1.8099192115795506, (30, 2) = .2708106376487236, (30, 3) = -.6614757705142081, (30, 4) = 0.3901692413302414e-1, (30, 5) = -0.9514173410140639e-1, (31, 1) = 1.8572174195820885, (31, 2) = .0, (31, 3) = -1.0333409329094383, (31, 4) = .0, (31, 5) = -.14906256152309377}, datatype = float[8], order = C_order); YP := Matrix(31, 5, {(1, 1) = .0, (1, 2) = -.9315092250220919, (1, 3) = 1.0, (1, 4) = -.34648123848818996, (1, 5) = -.0, (2, 1) = -.25157174296412993, (2, 2) = -.6266429386440938, (2, 3) = .876523695126908, (2, 4) = -.3452245048768078, (2, 5) = 0.11253962996950678e-1, (3, 1) = -.41143505643517725, (3, 2) = -.3644299174643001, (3, 3) = .7338541401987745, (3, 4) = -.3373854265507104, (3, 5) = 0.38857457998595e-1, (4, 1) = -.4939285418533152, (4, 2) = -.14996847236693056, (4, 3) = .5812930629123886, (4, 4) = -.31924318469866025, (4, 5) = 0.7255666249477757e-1, (5, 1) = -.5146039711733079, (5, 2) = 0.15286461167066837e-1, (5, 3) = .4314957440638574, (5, 4) = -.29044644794066193, (5, 5) = .10223647194417093, (6, 1) = -.4889090046306086, (6, 2) = .134080691208861, (6, 3) = .2965852977752991, (6, 4) = -.25353092284001993, (6, 5) = .12072431220789268, (7, 1) = -.430798678299882, (7, 2) = .21263948408802658, (7, 3) = .18482279369224633, (7, 4) = -.212609847242516, (7, 5) = .12550358382907195, (8, 1) = -.35198208297484934, (8, 2) = .2589494824064258, (8, 3) = 0.995186961141715e-1, (8, 4) = -.17198796501957242, (8, 5) = .11828756412983849, (9, 1) = -.26157835898450443, (9, 2) = .2813845520480499, (9, 3) = 0.39338927905785415e-1, (9, 4) = -.1350057879520591, (9, 5) = .10320388101597193, (10, 1) = -.16617020973083912, (10, 2) = .287445472379921, (10, 3) = 0.14968835622131138e-5, (10, 4) = -.10356868593432067, (10, 5) = 0.8472796647946662e-1, (11, 1) = -0.70328056399142e-1, (11, 2) = .28309382614500034, (11, 3) = -0.23817878855891178e-1, (11, 4) = -0.7828607488757705e-1, (11, 5) = 0.66369431369349e-1, (12, 1) = 0.23159682629474765e-1, (12, 2) = .2726118845713157, (12, 3) = -0.3720284228274004e-1, (12, 4) = -0.5878328825796292e-1, (12, 5) = 0.5018035246298953e-1, (13, 1) = .11272089234083588, (13, 2) = .2587658127130804, (13, 3) = -0.4421964517427246e-1, (13, 4) = -0.4419377987848922e-1, (13, 5) = 0.3696313433204147e-1, (14, 1) = .1974998734523982, (14, 2) = .24316815314012719, (14, 3) = -0.47768773248789226e-1, (14, 4) = -0.33522031258804846e-1, (14, 5) = 0.2671684617610927e-1, (15, 1) = .2769182605675772, (15, 2) = .226662356776616, (15, 3) = -0.4975761814895205e-1, (15, 4) = -0.25853335713280336e-1, (15, 5) = 0.19047304790426028e-1, (16, 1) = .35063241699188924, (16, 2) = .2095785407979796, (16, 3) = -0.5137458081352858e-1, (16, 4) = -0.20418369785624606e-1, (16, 5) = 0.13417084295042863e-1, (17, 1) = .4184566810307239, (17, 2) = .1919106490845611, (17, 3) = -0.53342768772377425e-1, (17, 4) = -0.16617772409198768e-1, (17, 5) = 0.929875139803741e-2, (18, 1) = .48016806829400327, (18, 2) = .173457068311107, (18, 3) = -0.56127906517646264e-1, (18, 4) = -0.14017723407396463e-1, (18, 5) = 0.6246987011637243e-2, (19, 1) = .5354415592106987, (19, 2) = .1538838140196576, (19, 3) = -0.6009502869660055e-1, (19, 4) = -0.12320225018360472e-1, (19, 5) = 0.3905246566824251e-2, (20, 1) = .5838153588501531, (20, 2) = .1327386247119349, (20, 3) = -0.6562847591727367e-1, (20, 4) = -0.11335102191701743e-1, (20, 5) = 0.19871265996410765e-2, (21, 1) = .6246747020405069, (21, 2) = .1094150967785294, (21, 3) = -0.7324576238712463e-1, (21, 4) = -0.10960526694470533e-1, (21, 5) = 0.2463596725038247e-3, (22, 1) = .6571695251212902, (22, 2) = 0.8309377267737389e-1, (22, 3) = -0.8373020420480838e-1, (22, 4) = -0.11175927429701141e-1, (22, 5) = -0.155897533952616e-2, (23, 1) = .6801269015815865, (23, 2) = 0.5261941106453635e-1, (23, 3) = -0.9833090510625761e-1, (23, 4) = -0.12048085031021012e-1, (23, 5) = -0.37151283448780886e-2, (24, 1) = .691903987919029, (24, 2) = 0.16281531791945737e-1, (24, 3) = -.11910534570976841, (24, 4) = -0.13757841466591127e-1, (24, 5) = -0.662562632759185e-2, (25, 1) = .6901321359389782, (25, 2) = -0.2852263328788306e-1, (25, 3) = -.14950637761935143, (25, 4) = -0.16656662943528553e-1, (25, 5) = -0.10928348172977546e-1, (26, 1) = .6713342300948061, (26, 2) = -0.859299780498995e-1, (26, 3) = -.19547726863609255, (26, 4) = -0.21376438602711868e-1, (26, 5) = -0.17700781115858367e-1, (27, 1) = .6302642389051082, (27, 2) = -.16271243923470902, (27, 3) = -.2674413824655153, (27, 4) = -0.29026728270966817e-1, (27, 5) = -0.28808435527763174e-1, (28, 1) = .559241686161435, (28, 2) = -.2696145103174367, (28, 3) = -.38318887606122914, (28, 4) = -0.4145590669149437e-1, (28, 5) = -0.473419350765662e-1, (29, 1) = .4461023878388593, (29, 2) = -.42517586122009443, (29, 3) = -.5742082155715147, (29, 4) = -0.6179362826521952e-1, (29, 5) = -0.7833603565269431e-1, (30, 1) = .2708106376487236, (30, 2) = -.6614757705142081, (30, 3) = -.8955655553848684, (30, 4) = -0.9514173410140639e-1, (30, 5) = -.12914913927984653, (31, 1) = .0, (31, 2) = -1.0333409329094383, (31, 3) = -1.4393540857249612, (31, 4) = -.14906256152309377, (31, 5) = -.20763118940116243}, 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(31, {(1) = .0, (2) = .3243000540262205, (3) = .6494459686383482, (4) = .9754443972533966, (5) = 1.3023020809731753, (6) = 1.6301555741550084, (7) = 1.9595249872617806, (8) = 2.2904837219758925, (9) = 2.623055009854351, (10) = 2.957225069568361, (11) = 3.292416584893846, (12) = 3.6284306063278815, (13) = 3.965273218644352, (14) = 4.302949344760717, (15) = 4.640942032851813, (16) = 4.978823623378006, (17) = 5.316594225819793, (18) = 5.654253949478462, (19) = 5.991736378579274, (20) = 6.328914836969517, (21) = 6.6657870456550175, (22) = 7.002353837596869, (23) = 7.338652555245001, (24) = 7.6748437956823565, (25) = 8.0109464436098, (26) = 8.346960569027788, (27) = 8.682730041734624, (28) = 9.016229528103496, (29) = 9.346894111650576, (30) = 9.674794880368847, (31) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(31, 5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = -0.34825708688301405e-7, (1, 4) = .0, (1, 5) = 0.8868979595878197e-8, (2, 1) = -0.173808460071561e-7, (2, 2) = 0.311777698147722e-8, (2, 3) = -0.27194031432912225e-7, (2, 4) = -0.9824058060923904e-9, (2, 5) = 0.9349182037774579e-8, (3, 1) = -0.2409380901812543e-7, (3, 2) = 0.7680735831101686e-8, (3, 3) = -0.2532269936410421e-7, (3, 4) = -0.4465307873187802e-9, (3, 5) = 0.1531437266543784e-7, (4, 1) = -0.20900093090721906e-7, (4, 2) = 0.7273301161705715e-8, (4, 3) = -0.28649116333750272e-7, (4, 4) = 0.6122579692376123e-8, (4, 5) = 0.21179868088379874e-7, (5, 1) = -0.13404639636134898e-7, (5, 2) = -0.18851926775553696e-8, (5, 3) = -0.29053132170340928e-7, (5, 4) = 0.18427473957834027e-7, (5, 5) = 0.20616569493780888e-7, (6, 1) = -0.9173055871683689e-8, (6, 2) = -0.16365941851348723e-7, (6, 3) = -0.20661986374008472e-7, (6, 4) = 0.3095868450296407e-7, (6, 5) = 0.13308354193228188e-7, (7, 1) = -0.12807375641681817e-7, (7, 2) = -0.2851387680120227e-7, (7, 3) = -0.6052793671111494e-8, (7, 4) = 0.3806295966709529e-7, (7, 5) = 0.5372449695184996e-8, (8, 1) = -0.23745716508846336e-7, (8, 2) = -0.3269314506944389e-7, (8, 3) = 0.7105149480768337e-8, (8, 4) = 0.3868851903947585e-7, (8, 5) = 0.25664978411467135e-8, (9, 1) = -0.3776879370565728e-7, (9, 2) = -0.28682574099694606e-7, (9, 3) = 0.13526428503011339e-7, (9, 4) = 0.3617053636720401e-7, (9, 5) = 0.5119726387441503e-8, (10, 1) = -0.5028248149002426e-7, (10, 2) = -0.2011708093503847e-7, (10, 3) = 0.13635641196804302e-7, (10, 4) = 0.3444652305244074e-7, (10, 5) = 0.93412187401973e-8, (11, 1) = -0.5852012120095596e-7, (11, 2) = -0.10984850820126323e-7, (11, 3) = 0.11295838579861379e-7, (11, 4) = 0.35320002993130185e-7, (11, 5) = 0.11923494660243025e-7, (12, 1) = -0.6194286264408458e-7, (12, 2) = -0.33144245982012917e-8, (12, 3) = 0.100910725255603e-7, (12, 4) = 0.38370969218624584e-7, (12, 5) = 0.12006686705032156e-7, (13, 1) = -0.6124629664737763e-7, (13, 2) = 0.290375217448951e-8, (13, 3) = 0.11536920982529403e-7, (13, 4) = 0.4233316807762056e-7, (13, 5) = 0.10420344191595053e-7, (14, 1) = -0.5730642217877111e-7, (14, 2) = 0.8697842512852038e-8, (14, 3) = 0.15427197256455848e-7, (14, 4) = 0.4616483571998453e-7, (14, 5) = 0.828967946754736e-8, (15, 1) = -0.50642151158086957e-7, (15, 2) = 0.1518820781808815e-7, (15, 3) = 0.208208180499055e-7, (15, 4) = 0.4935441333014615e-7, (15, 5) = 0.6333389210615828e-8, (16, 1) = -0.41302646372241204e-7, (16, 2) = 0.2316947743758456e-7, (16, 3) = 0.2674252751594911e-7, (16, 4) = 0.51813432997420677e-7, (16, 5) = 0.4799963236220735e-8, (17, 1) = -0.28945028254041156e-7, (17, 2) = 0.33030332030740915e-7, (17, 3) = 0.3248778269815313e-7, (17, 4) = 0.53659551283672146e-7, (17, 5) = 0.3663239899456364e-8, (18, 1) = -0.13012436352846456e-7, (18, 2) = 0.4483906962426248e-7, (18, 3) = 0.376236173112582e-7, (18, 4) = 0.5505157653362304e-7, (18, 5) = 0.28103066598690515e-8, (19, 1) = 0.7118389253789612e-8, (19, 2) = 0.5846772571389823e-7, (19, 3) = 0.4188559202557165e-7, (19, 4) = 0.56122737176541444e-7, (19, 5) = 0.21292216789364052e-8, (20, 1) = 0.3204340813553431e-7, (20, 2) = 0.7368168675733942e-7, (20, 3) = 0.4508192575717071e-7, (20, 4) = 0.5696649385493252e-7, (20, 5) = 0.15319349023142677e-8, (21, 1) = 0.6229496835797093e-7, (21, 2) = 0.902001930168359e-7, (21, 3) = 0.47031278545638364e-7, (21, 4) = 0.5764245711528697e-7, (21, 5) = 0.9497443963781525e-9, (22, 1) = 0.9830475940605532e-7, (22, 2) = 0.1077156077690418e-6, (22, 3) = 0.4752088065816434e-7, (22, 4) = 0.5818379486303611e-7, (22, 5) = 0.3210894950322663e-9, (23, 1) = 0.140379144652404e-6, (23, 2) = 0.12589832150568818e-6, (23, 3) = 0.4628721107453845e-7, (23, 4) = 0.58600360619335884e-7, (23, 5) = -0.42380157334763917e-9, (24, 1) = 0.1886617246186769e-6, (24, 2) = 0.14438010941710446e-6, (24, 3) = 0.4301293497044906e-7, (24, 4) = 0.58869838520755755e-7, (24, 5) = -0.1386555847493508e-8, (25, 1) = 0.2429799964624802e-6, (25, 2) = 0.1626605055663782e-6, (25, 3) = 0.37298027418436506e-7, (25, 4) = 0.5890404078521455e-7, (25, 5) = -0.2753549984560705e-8, (26, 1) = 0.3025730695147144e-6, (26, 2) = 0.1798759684261564e-6, (26, 3) = 0.2854072545734221e-7, (26, 4) = 0.5846146266868615e-7, (26, 5) = -0.4920411927579208e-8, (27, 1) = 0.36540121976197883e-6, (27, 2) = 0.19414923062428034e-6, (27, 3) = 0.15405061429325006e-7, (27, 4) = 0.56939925912625814e-7, (27, 5) = -0.8770161111958221e-8, (28, 1) = 0.42665573794613795e-6, (28, 2) = 0.20144720233204516e-6, (28, 3) = -0.56770357519229525e-8, (28, 4) = 0.53024431579258225e-7, (28, 5) = -0.1606596171900264e-7, (29, 1) = 0.4758832098400082e-6, (29, 2) = 0.1920698886768212e-6, (29, 3) = -0.4367930137685539e-7, (29, 4) = 0.4393075210907403e-7, (29, 5) = -0.2968630317846735e-7, (30, 1) = 0.4880309604521548e-6, (30, 2) = 0.14223953745304383e-6, (30, 3) = -0.11896192640699684e-6, (30, 4) = 0.25193702357403158e-7, (30, 5) = -0.5047080909285089e-7, (31, 1) = 0.40201550586535517e-6, (31, 2) = .0, (31, 3) = -0.2573217616510615e-6, (31, 4) = .0, (31, 5) = -0.5234847892716721e-7}, 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[31] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(4.880309604521548e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [5, 31, [F(eta), diff(F(eta), eta), diff(diff(F(eta), eta), eta), theta(eta), diff(theta(eta), eta)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[31] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[31] 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`(31, 5, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(5, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(31, 5, X, Y, outpoint, yout, L, V) end if; [eta = outpoint, seq('[F(eta), diff(F(eta), eta), diff(diff(F(eta), eta), eta), theta(eta), diff(theta(eta), eta)]'[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[31] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(4.880309604521548e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [5, 31, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[31] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[31] 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`(31, 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`(31, 5, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 5)] end proc, (2) = Array(0..0, {}), (3) = [eta, F(eta), diff(F(eta), eta), diff(diff(F(eta), eta), eta), theta(eta), diff(theta(eta), eta)], (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); [eta = res[1], seq('[F(eta), diff(F(eta), eta), diff(diff(F(eta), eta), eta), theta(eta), diff(theta(eta), eta)]'[i] = res[i+1], i = 1 .. 5)] catch: error  end try end proc

 

 

 

Download odeProb.mw

See the attached, where I have used arbitrary values for alpha, c, gamma and I am assuming that you are only interested in real roots

restart

rel := w+(1/2)*sqrt(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)

w+(1/2)*(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)^(1/2)

(1)

#
# Arbitrary values for alpha, c, gamma
#
  pvals:=[alpha=1, c=2, gamma=3];
  poly:=eval( numer(diff(rel, k)), pvals);
#
# Get all roots of poly including complex ones
#
  [fsolve(poly, complex)];
#
# Select only real roots
#
  crit_points:=select( type, %, realcons);

[alpha = 1, c = 2, gamma = 3]

 

-72*k^7+42*k^6+3468*k^5-245*k^4-49057*k^3-10416*k^2+278408*k+27840

 

[HFloat(-5.1603945076304925), -HFloat(3.1908916702688197)-HFloat(1.2540904681468148)*I, -HFloat(3.1908916702688197)+HFloat(1.2540904681468148)*I, HFloat(-0.09979943545723942), HFloat(3.2293215429038007)-HFloat(0.8048613105794764)*I, HFloat(3.2293215429038007)+HFloat(0.8048613105794764)*I, HFloat(5.7666675311511035)]

 

[HFloat(-5.1603945076304925), HFloat(-0.09979943545723942), HFloat(5.7666675311511035)]

(2)

for i to nops(crit_points) do rel_at_crit[i] := eval(rel, [pvals[], k = crit_points[i]]); if eval(diff(diff(rel, k), k), [pvals[], k = crit_points[i]]) > 0 then print("Minimum at k = ", crit_points[i], " with value ", rel_at_crit[i]) elif eval(diff(diff(rel, k), k), [pvals[], k = crit_points[i]]) < 0 then print("Maximum at k = ", crit_points[i], " with value ", rel_at_crit[i]) else print("Saddle point at k = ", crit_points[i], " with value ", rel_at_crit[i]) end if end do

w+HFloat(1151.9956598109318)

 

"Maximum at k = ", HFloat(-5.1603945076304925), " with value ", w+HFloat(1151.9956598109318)

 

w+HFloat(37.27893466430185)*I

 

"Saddle point at k = ", HFloat(-0.09979943545723942), " with value ", w+HFloat(37.27893466430185)*I

 

w+HFloat(1330.6994712946923)

 

"Maximum at k = ", HFloat(5.7666675311511035), " with value ", w+HFloat(1330.6994712946923)

(3)

NULL

Download crits.mw

If you examine the numerator of diff(rel, k), which has to be zero to get the critical points, you will find that it is 7-th order polynomial in k. Only polynomials of order 4 or less can be solved analytically - so your solve() commands will produce nothing useful.

If you have numerical values for {alpha, c, gamma, w}, tyhen you can use fsolve()

that

most of papers or books give directly the formula without any proof. 

because as https://en.wikipedia.org/wiki/Characteristic_equation_(calculus) states

In mathematics, the characteristic equation (or auxiliary equation[1]) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation[2] or difference equation.[3][4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients.

Note that the characteristic equation can only be found when the differential equation is linear and you specifically state

I have a system of nonlinear differential equation

 

ie '%/', as in the attached

  restart;
#
# Evaluates the argument before generating latex
#
  latex((4*n-1)/9-7/16*n);
#
# Use the inert form of '/'
#
  latex((4*n-1)%/9-7%/16*n);
  

\frac{n}{144}-\frac{1}{9}
\frac{4 n -1}{9}-\frac{7}{16} n

 

 

Download inertDiv.mw

once again no uploaded worksheet.

The " code" which you present has obviously been cut-and-pasted from a workseet using 2D input. It has multiple instances where the 2D input is interpreting spaces as implicit multiplication. Consider the snips from your (incorrect) code, with the highlighted multiplication signs below

display*([draw*[A(color = black, symbol = solidcircle, symbolsize = 6), 
B(color = black, symbol = solidcircle, symbolsize = 6), 
C(color = black, symbol = solidcircle, symbolsize = 6), 
ABC(color = blue)], 
textplot*([[coordinates(A)[], "A"], 
[coordinates(B)[], "B"], 
[coordinates(C)[], "C3"]], 
align = [above, right])], 
axes = none, 
title = "Lemme du Chevron");

There are also multiple unbalanced parentheses.

I have fixed all of these in the attached.

restart;
with(geometry):
with(plots):
_EnvHorizontalName = 'x':
_EnvVerticalName = 'y':
EQ := proc(M, N)
local eq, sol;
eq := simplify(expand((y - M[2])/(x - M[1]) - (N[2] - M[2])/(N[1] - M[1])));
sol := solve(eq, y);
RETURN(y = sol); end proc:
_local(D):
point(A, [-2, 7]):
point(B, [-5, -2]):
point(C, [8, -7]):
point(E, [1, 4]):
EQ([-5, -2], [8, -7]):
point(D, [1, subs(x = 1, rhs(%))]):
dsegment(sgAD, [A, D]):
BD := distance(B, D):
DC := distance(C, D):
triangle(ABC, [A, B, C]):
area(ABC):
triangle(ABD, [A, B, D]):
area(ABD):
triangle(ADC, [A, D, C]):
area(ADC):
is(area(ABD)/area(ADC) = BD/DC):
triangle(EBD, [E, B, D]):
area(EBD):
triangle(EDC, [E, D, C]):
area(EDC):
triangle(AEC, [A, E, C]):
area(AEC):
triangle(ABE, [A, B, E]):
area(ABE):
is(area(ABE)/area(AEC) = BD/DC):
display
( [ draw
    ( [ A(color = black, symbol = solidcircle, symbolsize = 6),
        B(color = black, symbol = solidcircle, symbolsize = 6),
        C(color = black, symbol = solidcircle, symbolsize = 6),
        ABC(color = blue)
      ]
    ),
    textplot
    ( [ [coordinates(A)[], "A"],
        [coordinates(B)[], "B"],
        [coordinates(C)[], "C3"]
      ],
      align = [above, right]
    )
  ],
  axes = none,
  title = "Lemme du Chevron"
);

 

 

Download chev.mw

makes a fairly trivial calculation look really complicated. In the attached, I deleted absolutely everything that you do not need. So much simpler to understand what is happening!

  restart:
  with(plots):
  Params := [ fw = 0.2, M = 0.5, Q = 0.5, Pr = 6.2, phi = 0.05,
              rf = 997.1, kf = 0.613, cpf = 4179, btf = 0.00003,
              p1 = 0.01, p2 = 0.05, p3 = 0.05, rs1 = 5100, ks1 = 3007.4,
              cps1 = 410, bs1 = 0.0002, rs2 = 2200, ks2 = 5000, cps2 = 790,
              bs2 = 0.0005, rs3 = 3970, ks3 = 40, cps3 = 765, bs3 = 0.0004,
              A1 = B1*p1 + B2*p2 + B3*p3, B1 = 1 + 2.5*phi + 6.2*phi^2,
              B2 = 1 + 13.5*phi + 904.4*phi^2, B3 = 1 + 37.1*phi + 612.6*phi^2,
              B4 = (ks1 + 2*kf - 2*phi*(kf - ks1))/(ks1 + 2*kf + phi*(kf - ks1)),
              B5 = (ks2 + 3.9*kf - 3.9*phi*(kf - ks2))/(ks2 + 3.9*kf + phi*(kf - ks2)),
              B6 = (ks3 + 4.7*kf - 4.7*phi*(kf - ks3))/(ks3 + 4.7*kf + phi*(kf - ks3)),
              A2 = 1 - p1 - p2 - p3 + p1*rs1/rf + p2*rs2/rf + p3*rs3/rf,
              A3 = B4*p1 + B5*p2 + B6*p3, A4 = 1 - p1 - p2 - p3 + p1*rs1*cps1/(rf*cpf) + p2*rs2*cps2/(rf*cpf) + p3*rs3*cps3/(rf*cpf)
          ]:
  ODEs:= [ A1*(diff(f(x), x, x, x))/(A2*phi)-(diff(f(x), x))^2-M^2*(f(x))+f(x)*(diff(f(x), x, x)),
           A4*Pr*phi*(diff(Theta(x), x, x))/A3+f(x)*(diff(Theta(x), x))+Q*Theta(x)
         ]:
  ODEs:=eval[recurse](ODEs, Params):
  BCs := [f(0) = fw, D(f)(0) = 1, Theta(0) = 1, D(f)(1) = 0, Theta(1) = 0]:
  BCs := eval[recurse](BCs,Params):
  sol:=dsolve([ODEs[], BCs[]], numeric);
  odeplot( sol,
           [ [ x, f(x)],
             [x, Theta(x)]
           ],
           x=0..1,
           color=[red, blue]
         );

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(8, {(1) = .0, (2) = .13834908494511125, (3) = .2780635977051906, (4) = .4198594041085766, (5) = .5637966858285377, (6) = .7096694087269098, (7) = .8571389727881707, (8) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(8, 5, {(1, 1) = 1.0, (1, 2) = -1.0295000678638335, (1, 3) = .2, (1, 4) = 1.0, (1, 5) = -1.0767337173480696, (2, 1) = .856569909795754, (2, 2) = -1.0417108589688173, (2, 3) = .32812427303302627, (2, 4) = .8527522204005735, (2, 5) = -1.0522812834212694, (3, 1) = .7109259541331585, (3, 2) = -1.0411403186341557, (3, 3) = .43707085930864015, (3, 4) = .7073340848489478, (3, 5) = -1.0296885843248846, (4, 1) = .564028949899526, (4, 2) = -1.0290320485455207, (4, 3) = .5270885296373472, (4, 4) = .5628455479836898, (4, 5) = -1.0085664552506046, (5, 1) = .417402343699789, (5, 2) = -1.006845007833673, (5, 3) = .5977252490881858, (5, 4) = .4191257730135991, (5, 5) = -.9886363213565754, (6, 1) = .2726733911493102, (6, 2) = -.9762939076067769, (6, 3) = .6484140152858878, (6, 4) = .2763086891518715, (6, 5) = -.9696427654924297, (7, 1) = .13136719462469473, (7, 2) = -.939253672846738, (7, 3) = .6786846128471948, (7, 4) = .1346750596424013, (7, 5) = -.9513307699360147, (8, 1) = .0, (8, 2) = -.8993106332471071, (8, 3) = .6882752611704436, (8, 4) = .0, (8, 5) = -.9341330432755275}, datatype = float[8], order = C_order); YP := Matrix(8, 5, {(1, 1) = -1.0295000678638335, (1, 2) = -.13766016841596407, (1, 3) = 1.0, (1, 4) = -1.0767337173480696, (1, 5) = .18505997004013197, (2, 1) = -1.0417108589688173, (2, 2) = -0.4047627435482859e-1, (2, 3) = .8527522204005735, (2, 4) = -1.0522812834212694, (2, 5) = .1688478508598343, (3, 1) = -1.0411403186341557, (3, 2) = 0.4661494459186713e-1, (3, 3) = .7073340848489478, (3, 4) = -1.0296885843248846, (3, 5) = .1549742988133227, (4, 1) = -1.0290320485455207, (4, 2) = .12187513256505261, (4, 3) = .5628455479836898, (4, 4) = -1.0085664552506046, (4, 5) = .14335234723749268, (5, 1) = -1.006845007833673, (5, 2) = .18400663034061815, (5, 3) = .4191257730135991, (5, 4) = -.9886363213565754, (5, 5) = .1339716559681159, (6, 1) = -.9762939076067769, (6, 2) = .232494487830241, (6, 3) = .2763086891518715, (6, 4) = -.9696427654924297, (6, 5) = .12682713036976728, (7, 1) = -.939253672846738, (7, 2) = .2676315153056201, (7, 3) = .1346750596424013, (7, 4) = -.9513307699360147, (7, 5) = .12189581107682387, (8, 1) = -.8993106332471071, (8, 2) = .2897244722294419, (8, 3) = .0, (8, 4) = -.9341330432755275, (8, 5) = .1191970743667628}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(8, {(1) = .0, (2) = .13834908494511125, (3) = .2780635977051906, (4) = .4198594041085766, (5) = .5637966858285377, (6) = .7096694087269098, (7) = .8571389727881707, (8) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(8, 5, {(1, 1) = .0, (1, 2) = -0.16313919815053183e-6, (1, 3) = .0, (1, 4) = .0, (1, 5) = 0.31121062393797565e-8, (2, 1) = -0.8754717109590983e-7, (2, 2) = -0.16412070890656207e-6, (2, 3) = 0.206430400648348e-7, (2, 4) = -0.19215829844036117e-8, (2, 5) = 0.809631728327652e-8, (3, 1) = -0.1391673972144928e-6, (3, 2) = -0.15640199167454757e-6, (3, 3) = 0.4127822669510205e-7, (3, 4) = -0.2847243457679656e-8, (3, 5) = 0.12385748243706898e-7, (4, 1) = -0.15526795764104e-6, (4, 2) = -0.14971077547680476e-6, (4, 3) = 0.6255431406201337e-7, (4, 4) = -0.3014229649221912e-8, (4, 5) = 0.16016555082231332e-7, (5, 1) = -0.13933078171207597e-6, (5, 2) = -0.15067077692184236e-6, (5, 3) = 0.8459096728921743e-7, (5, 4) = -0.26192201699297497e-8, (5, 5) = 0.18907966492537008e-7, (6, 1) = -0.9892914819005134e-7, (6, 2) = -0.16194069796824688e-6, (6, 3) = 0.1072576018668421e-6, (6, 4) = -0.18783484184799645e-8, (6, 5) = 0.20968637026752008e-7, (7, 1) = -0.443536611542425e-7, (7, 2) = -0.18217346881865962e-6, (7, 3) = 0.13026651060348097e-6, (7, 4) = -0.1035785886232914e-8, (7, 5) = 0.22138451230600717e-7, (8, 1) = .0, (8, 2) = -0.20192994539838603e-6, (8, 3) = 0.14917304794752758e-6, (8, 4) = .0, (8, 5) = 0.22575111546841176e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[8] elif outpoint = "order" then return 4 elif outpoint = "error" then return HFloat(2.0192994539838603e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [5, 8, [Theta(x), diff(Theta(x), x), f(x), diff(f(x), x), diff(diff(f(x), x), x)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[8] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[8] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(5, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(8, 5, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(5, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(8, 5, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[Theta(x), diff(Theta(x), x), f(x), diff(f(x), x), diff(diff(f(x), x), x)]'[i] = yout[i], i = 1 .. 5)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[8] elif outpoint = "order" then return 4 elif outpoint = "error" then return HFloat(2.0192994539838603e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [5, 8, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[8] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[8] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(8, 5, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false)  ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(5, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0.}); `dsolve/numeric/hermite`(8, 5, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 5)] end proc, (2) = Array(0..0, {}), (3) = [x, Theta(x), diff(Theta(x), x), f(x), diff(f(x), x), diff(diff(f(x), 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('[Theta(x), diff(Theta(x), x), f(x), diff(f(x), x), diff(diff(f(x), x), x)]'[i] = res[i+1], i = 1 .. 5)] catch: error  end try end proc

 

 

 

NULL

Download strangeODE.mw

at ?plot3d,coords. Note in particular the interpretation of the expression being plotted, which is stated as

For spherical coordinates the interpretation is:

"plot3d(r(theta, phi), theta = a .. b, phi = c .. d, coords =spherical)"

So if you want to plot a sphere of radius 1, you would use something like the attached

(By the way, there are many other ways to plot a sphere!!)

  restart;
  plot3d( 1,
          theta = 0..Pi,
          phi = 0..2*Pi,
          coords = spherical,
          style =surface,
          scaling=constrained
        );

 

 

Download sphPlot.mw

 

 

with generating a conic from five points, although I would probably let the the command conic() in Maple's geometry() package do most of the "work" - as in the attached.

  restart:
  with(geometry):
  
  Pn := [ (5*x + 12*y)*(-384 + 48*x + 55*y)/(225*x^2 + 225*y^2 - 1800*x - 602*y),
          2*(12*x - 5*y)*(-384 + 48*x + 55*y)/(225*x^2 + 225*y^2 - 1800*x - 602*y)
        ]:
  line(l1, 2+3*x-y, [x,y]):
#
# Define the conic from five points
#
  conic( c,
         [ seq
           ( point
             ( cat(P__, j),
               eval
               ( eval( Pn, isolate( Equation(l1), y ) ),
                 x=j
               )
             ),
             j=1..5
           )
         ],
         [x,y]
        ):
#
# Some properties of the conic
#
  form(c);
  simplify(coordinates(center(c)));
  simplify~(coordinates~(foci(c)));
  simplify(Equation(c));
#
# Draw the conic
#
  c_cen:=coordinates(center(c)):
  vr:= z-> [ c_cen[1]-z..c_cen[1]+z,
             c_cen[2]-z..c_cen[2]+z
           ]:
  draw( c(color=red),
        view=vr(30)
      );

hyperbola2d

 

[1123561/98801, -640748/98801]

 

[[1123561/98801+(949/233520218194239)*(13^(1/4)*369361^(3/4)+(502/13)*13^(3/4)*369361^(1/4))*82^(1/2)*(46112511332498-4820899772*4801693^(1/2))^(1/2), -(73/109479708483)*82^(1/2)*369361^(1/4)*13^(3/4)*(46112511332498-4820899772*4801693^(1/2))^(1/2)-640748/98801], [1123561/98801-(949/233520218194239)*(13^(1/4)*369361^(3/4)+(502/13)*13^(3/4)*369361^(1/4))*82^(1/2)*(46112511332498-4820899772*4801693^(1/2))^(1/2), (73/109479708483)*82^(1/2)*369361^(1/4)*13^(3/4)*(46112511332498-4820899772*4801693^(1/2))^(1/2)-640748/98801]]

 

-(297332841114360453337351533/87612171760745984318337651200)*x^2+(1/175224343521491968636675302400)*(-1718728862051303108315910081*y+2378662728914883626698812264)*x-(24979181773835160036471267/4380608588037299215916882560)*y*(y-8173/1240) = 0

 

 

 

Download hyp.mw

if you ha uploaded a worksheet using the big green up-arrow in the Mapleprimes toolbar,

Taking a wild guess, try replacing

YFtpt5 := TtT -> solution(TtT, 0.5)[3]

with

YFtpt5 := TtT -> rhs(solution(TtT, 0.5)[3])

extract data from a plot, when it is simpler to generate the data used by the plot?

See the final execution group in the attached where I have arrabged the the data matrix is in the same "orientation" as the contour plot, just to make comparing the two a bit simpler.

data.mw

As well as fixing syntax problems, I had to make many wild guesses about what you are actually trying to achieve.

Maybe(??!) the attached will be helpful

  restart:
  N := 4: M := .1: Kp := .1: Gr := 0.1e-1: Gc := 0.1e-1: Pr := 1: S := 0.1e-1: Sc := .78: Kc := 0.1e-1: La := 1:
  f__N:=  sum((p^(i))*f[i](x), i = 0 .. N) :
  Theta__N:=  sum((p^(i))*Theta[i](x), i = 0 .. N) :
  Phi__N:= sum((p^(i))*Phi [i] (x), i = 0 .. N):
  HPMEq1 := (1-p)*(diff(f__N, x, x, x))+p*(diff(f__N, x, x, x)+(1/2)*(diff(f__N, x, x))*f__N-(M^2+Kp)*(diff(f__N, x)-La)+Gr*Theta__N+Gc*Phi__N):
  HPMEq2 := (1-p)*(diff(Theta__N, x, x))/Pr+p*((diff(Theta__N, x, x))/Pr+(1/2)*(diff(Theta__N, x))*f__N+S*Theta__N):
  HPMEq3 := (1-p)*(diff(Phi__N, x, x))/Sc+p*((diff(Phi__N, x, x))/Sc+(1/2)*(diff(Phi__N, x))*f__N+Kc*Phi__N):

  for i from 0 to N do
      equ[1][i] := coeff(HPMEq1, p, i) = 0:
  end do:
  for i from 0 to N do
      equ[2][i] := coeff(HPMEq2, p, i) = 0:
  end do:
  for i from 0 to N do
      equ[3][i] := coeff(HPMEq3, p, i) = 0:
  end do:
  cond[1][0] := f[0](0) = 0, (D(f[0]))(0) = 0, Theta[0](0) = 1, Phi[0](0) = 1, Theta[0](5) = 0, Phi[0](5) = 0, (D(f[0]))(5) = 1:
  for j to N do
       cond[1][j] := f[j](0) = 0, (D(f[j]))(0) = 0, Theta[j](0) = 0, Phi[j](0) = 0, Theta[j](5) = 0, Phi[j](5) = 0, (D(f[j]))(5) = 0:
  end do:

  ans:=dsolve({equ[1][0], equ[2][0], equ[3][0], cond[1][0]}):
  for k from 1 by 1 to 4 do
      ans:=`union`( ans, dsolve( { eval({equ[1][k], equ[2][k], equ[3][k]}, ans)[], cond[1][k]})):
  od:
  Phi:= evalf(eval(Phi[k-1](x), ans));
  Theta:= evalf(eval(Theta[k-1](x), ans));
  f:= evalf(eval(f[k-1](x), ans));
  plot([Phi, Theta, f], x=0..5, color=[red, blue, green]);

-0.3896273449e-10*x^13+0.1697586667e-9*x^12-0.4688867837e-8*x^11+0.5906981367e-7*x^10-0.1174292121e-6*x^9+0.1327469799e-5*x^8-0.1074474492e-4*x^7+0.1506166234e-5*x^6-0.3446424640e-4*x^5+0.2912878380e-3*x^4-0.1106746846e-4*x^3+0.2239032281e-1*x

 

-0.8959982918e-10*x^13+0.3158683562e-9*x^12-0.8909098531e-8*x^11+0.1145236794e-6*x^10-0.1741583958e-6*x^9+0.2150994486e-5*x^8-0.1902330877e-4*x^7+0.1921094746e-5*x^6-0.5632953871e-4*x^5+0.4909768121e-3*x^4-0.2416158293e-4*x^3+0.3543341013e-1*x

 

0.6399987799e-11*x^14-0.7215804519e-10*x^13+0.1619819726e-8*x^12-0.2016116095e-7*x^11+0.1472981866e-6*x^10-0.1207325236e-5*x^9+0.5883077911e-5*x^8-0.9303180532e-5*x^7-0.3315385078e-5*x^6+0.6960558200e-4*x^5-0.1315517192e-3*x^4-0.4510232982e-2*x^2

 

 

 

Download ODEnotPDE.mw

 

Consider the command

x:=x+1

If 'x' has previously been assigned a numeric value, the this will work as expected. But if 'x' has not been assigned a numeric value, then the above will evaluate (recursively) to

(x+1)+1;
((x+1)+1)+1
(((x+1)+1)+1)+1....

and so on. After a sufficient number of such recursive evaluations, Maple will halt and generate an error.

Almost any assignment can produce this. Unless you post a worksheet, using the big green up-arrow in the Mapleprimes toolbar, no-one will be able to tell "which command created it" - although it is probaly related to an assignment to the quantity 'PD'

with RealDomain() - and a version issue.

Maple 2022.2, returns x=-4, x=2, as solutions for your equation, whether or not you invoke RealDomain(). When RealDomain() is not specified, both sides of your equation evaluate to the same (complex) number. However, if one attempts to evaluate your equation at x=2, after invoking RealDomain(), then Maple returns undefined - see the attached.

#
# No real domain assumption
#
  restart;
  eqn:= log[2](x^2 - 6*x) = 3 + log[2](1 - x);
  ans:=solve(eqn, x);
#
# x=2 gives the function complex values!
#
  simplify(eval(eqn, x=ans[1]));
  simplify(eval(eqn, x=ans[2]));

ln(x^2-6*x)/ln(2) = 3+ln(1-x)/ln(2)

 

2, -4

 

(3*ln(2)+I*Pi)/ln(2) = (3*ln(2)+I*Pi)/ln(2)

 

(3*ln(2)+ln(5))/ln(2) = 3+ln(5)/ln(2)

(1)

#
# Use real domain
#
  with(RealDomain):
  ans:=solve(eqn, x);
  simplify(eval(eqn, x=ans[1]));
  simplify(eval(eqn, x=ans[2]));

2, -4

 

undefined

 

(3*ln(2)+ln(5))/ln(2) = 3+ln(5)/ln(2)

(2)


Download realDom.mw

Apart from the restricted viewing ranges in the required plots (which I have removed in the attached), I don't see any problem with this worksheet

restartNULL

with(PDEtools); with(PolynomialIdeals); with(DifferentialGeometry); with(plots)

ODE1 := diff(f(x), x, x, x)+(1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[f(x)])*f(x)*(diff(f(x), x, x))+(1-phi)^2.5*M*`sin&alpha;`^2*(1-(diff(f(x), x)))+(1-phi)^2.5*(1-phi+phi*`&rho;&beta;`[s]/`&rho;&beta;`[f(x)])*Gr[x]*theta = 0

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

(1)

ODE2 := K[nf]*(diff(theta(x), x, x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*C[p][s]/(rho[f]*C[p][f]))*f*(diff(theta(x), x)) = 0

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

(2)

Implementation*of*Differential*Transformation*Method*(Fe[3]*O[4]-Water)

Implementation*of*Differential*DifferentialGeometry:-Transformation*Method*(Fe[3]*O[4]-Water)

(3)

"rho[s]:=5200:rho[f]:=997.1:(rhobeta)[s]:=6500:(rhobeta)[f]:=20939.1:(C[p])[s]:=670:(C[p])[f]:=4179:K[s]:=6:K[f]:=0.613: K[ nf]:=K[f]*(((K[s]+2*K[f])-2*phi*(K[f]-K[s]))/((K[s]+2*K[f])+phi*(K[f]-K[s]))): F(0):=0:F(1):=`&epsilon;`:F(2):=A:delta(-1):=0: delta(0):=1:delta(1):=0: for k from 2 to 20 do delta(k):=0: Theta(0):=0:Theta(1):=B:od:"

for k from 0 to 15 do F(k+3) := -((1/2)*(1-phi)^2.5*(1-phi+phi*rho[s]/rho[f])*(sum((m+1)*(m+2)*F(m+2)*F(k-m), m = 0 .. k))+(1-phi)^2.5*M*`sin&alpha;`^2*(1-(k+1)*F(k+1))+(1-phi)^2.5*(1-phi+phi*`&rho;&beta;`[s]/`&rho;&beta;`[f])*Gr[x]*Theta(k))/((k+1)*(k+2)*(k+3)); Theta(k+2) := -(1/2)*K[f]*Pr*(1-phi+phi*rho[s]*C[p][s]/(rho[f]*C[p][f]))*(sum((m+1)*Theta(m+1)*F(k-m), m = 0 .. k))/((k+1)*(k+2)*K[nf]) end do

"f(eta):=simplify(sum(F(i)*eta^(i),i=0..5),symbolic):"

indets(f(eta))

{A, B, M, epsilon, eta, phi, `sin&alpha;`, Gr[x], (1.-phi)^(1/2)}

(4)

"Theta(eta):=simplify(sum(Theta(i)*eta^(i),i=0..5),symbolic):"

indets(Theta(eta))

{A, B, M, Pr, epsilon, eta, phi, `sin&alpha;`, (1.-phi)^(1/2)}

(5)

U := subs(M = 1, phi = .1, `sin&alpha;` = 1, Gr[x] = .1, Pr = 6.2, `&epsilon;` = 0, `&rho;&beta;`[f] = 20939.1, K[nf] = .6842, f(eta))

-0.7025206432e-1*eta*(.9486832981*(.1921651717+.2731651717*A^2)*eta^4+.9486832981*(-.9608258583*A+.4804129292+0.4472848085e-1*B)*eta^3+1.823038889*eta^2+0.7004420503e-1*eta^4-14.23445716*A*eta)

(6)

V := subs(Pr = 6.2, phi = .1, `sin&alpha;` = 1, M = 1, Gr[x] = .1, `&epsilon;` = 0, `&rho;&beta;`[f] = 20939.1, K[nf] = .6842, Theta(eta))

-0.9677800978e-2*(-1.624907219*eta^4-103.3292586+21.14571110*A*eta^3)*B*eta

(7)

e4 := subs(eta = 7, diff(U, eta)) = 0; e5 := subs(eta = 7, V) = 0; j := {e4, e5}; j := evalf(solve(j))

-275.5797343-218.5585676*A^2+101.8575602*A-4.089955705*B = 0

 

-0.6774460685e-1*(-4004.731492+7252.978907*A)*B = 0

 

{-0.6774460685e-1*(-4004.731492+7252.978907*A)*B = 0, -275.5797343-218.5585676*A^2+101.8575602*A-4.089955705*B = 0}

 

{A = .5521498881, B = -69.92030121}, {A = .2330212019+1.098452378*I, B = 0.}, {A = .2330212019-1.098452378*I, B = 0.}

(8)

U1 := subs(A = .5521498883, B = -69.92030119, U)

-0.7025206432e-1*eta*(.3313541819*eta^4-3.014475156*eta^3+1.823038889*eta^2-7.859553931*eta)

(9)

V1 := subs(A = .5521498883, B = -69.92030119, V)

.6766747592*(-1.624907219*eta^4-103.3292586+11.67560202*eta^3)*eta

(10)

p1 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 0, `&epsilon;` = -.2, Gr[x] = .1, Pr = 6.2; p2 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 1/sqrt(2), `&epsilon;` = 0, Gr[x] = .1, Pr = 6.2; p3 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 1, `&epsilon;` = .2, Gr[x] = .1, Pr = 6.2

s1 := subs(p1, f(eta)); s2 := subs(p2, f(eta)); s3 := subs(p3, f(eta))

d1 := diff(s1, `$`(eta, 1)); d2 := diff(s2, `$`(eta, 1)); d3 := diff(s3, `$`(eta, 1))

plot([d1, d2, d3], eta = 0 .. 7)

 

p1 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 0, `&epsilon;` = -.2, Gr[x] = .1, Pr = 6.2; p2 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 1/sqrt(2), `&epsilon;` = 0, Gr[x] = .1, Pr = 6.2; p3 := M = 1, phi = .1, A = .5521498883*(1/2), B = -69.92030119, `sin&alpha;` = 1, `&epsilon;` = .2, Gr[x] = .1, Pr = 6.2

s1 := subs(p1, Theta(eta)); s2 := subs(p2, Theta(eta)); s3 := subs(p3, Theta(eta))

plot([s1, s2, s3], eta = 0 .. 7)

 

NULL

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