Kitonum

14069 Reputation

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11 years, 198 days

MaplePrimes Activity


These are answers submitted by Kitonum

Example:

combinat:-permute([a,b,c,d]);

       [[a, b, c, d], [a, b, d, c], [a, c, b, d], [a, c, d, b], [a, d, b, c], [a, d, c, b], [b, a, c, d], [b, a, d, c], [b, c, a, d], [b, c, d, a], [b, d, a, c], [b, d, c, a], [c, a, b, d], [c, a, d, b], [c, b, a, d], [c, b, d, a], [c, d, a, b], [c, d, b, a], [d, a, b, c], [d, a, c, b], [d, b, a, c], [d, b, c, a], [d, c, a, b], [d, c, b, a]] 

In fact, there are many more solutions than indicated by vv, for example

restart;
sys := [x^2 + y^2 - x*y - 1 = 0, y^2 + z^2 - y*z - a^2 = 0, z^2 + x^2 - x*z - b^2 = 0];
fsolve(eval(sys,[a=10,b=10.4]), {x=0..infinity,y=0..infinity,z=0..infinity});
;

                      sys := [x^2-x*y+y^2-1 = 0, -a^2+y^2-y*z+z^2 = 0, -b^2+x^2-x*z+z^2 = 0]
                                    {x =0.1989730189, y = 1.084528287, z = 10.49805888}


Further, we strictly show that there are positive solutions  (x, y, z)  for arbitrary a=b>=1. From considerations of continuity, this implies that for any number  a>=1 and a sufficiently small positive number c  for  (a, b=a+c)  there are also a positive solutions  (x, y, z) .

restart;
sys := [x^2 + y^2 - x*y - 1 = 0, y^2 + z^2 - y*z - a^2 = 0, z^2 + x^2 - x*z - b^2 = 0];
Sol:=simplify(solve(eval(sys,[b=a]), [x,y,z], explicit));
a in solve(a>=1 and `or`(seq((eval(x,k)>0 and eval(y,k)>0 and eval(z,k)>0), k=%)));

 The result:                                          a in RealRange(1, Open(infinity))



Numerical experiments show that all solutions seem to be in the strip  1<=a<=b<a+1/2 . This does not mean that every point of this strip is a solution. This is only a necessary condition.

restart;
a := 1: b:= 2: l:= 1: alpha[1]:= 1: alpha[2]:= 3:
syspde:= [diff(u(x, t), t)-a+u(x, t)-u(x, t)^2*v(x, t)-alpha[1]*(diff(u(x, t), x$2)) = 0, diff(v(x, t), t)-b+u(x, t)^2*v(x, t)-alpha[2]*(diff(v(x, t), x$2)) = 0]:
Bcs:= [u(x,0)=1,v(x,0)=1,D[1](u)(-l, t) = 0,D[1](u)(l, t) = 0,D[1](v)(-l, t) = 0,D[1](v)(l, t) = 0]:
sol:=pdsolve(syspde, Bcs, numeric);

sol:-value(u(x, t)+diff(u(x, t), x)+diff(u(x, t), t), t=0.7)(0.5);

                                          sol:=module() ... end module
               [x = 0.5, t = 0.7, u(x, t)+diff(u(x, t), x)+diff(u(x, t), t) = 2.34305224942915]

restart;	
s := Sum(a*X[n]+b, n=1..N);
expand(value(s));
op(2,%)%+op(1,%); 

s := Sum(X[n]+Y[n], n=1..N);
(expand@value)(s);

                                    

 

I use Maple 2018.2

Do you mean this:

Example:

for n from 1 to 10 do
a[n]:=n^2;
od:

seq(a[n], n=1..10);
                                 
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100

The reason for the lack of a solution can be not only a comma. You solve a linear system with 9 unknowns, but in the solve command you specify only 6 unknowns. Could well be a situation that the system is consistent, but the unknowns, that you specified, are not expressed in terms of the remaining unknowns. Here is a simple example:

solve({x-2*y+3*z=0,2*x-4*y-z=1},{x,y,z});
solve({x-2*y+3*z=0,2*x-4*y-z=1},{x,y});
solve({x-2*y+3*z=0,2*x-4*y-z=1},{x,z});

                    {x = 2*y+3/7,  y = y,  z = -1/7}
                                        NULL
                          {x = 2*y+3/7, z = -1/7}


In the first case, Maple returns a general solution. We see that there are infinitely many solutions (a straight line of the solutions). But in the second case Maple does not return anything (NULL), because {x,y} cannot be expressed through  z (the line is perpendicular to  Oz-axis). The third case works again.

The other two answers consider only simple examples in which one can easily find a parameterization or one variable is explicitly expressed through another. Here is a more complicated case in which these methods do not work. Here, we first plot the curve given by the implicit equation, then extract the data using  plottools:-data  command and finally do the animation using these data:


 

restart;

with(plots):
opt:=color=red,thickness=4:
implicitplot(x^5+y^7-3*x*y=0, x=-2..2,y=-2..2,opt, gridrefine=2);
data:=plottools:-getdata(%);
data1:=data[1,-1]:
data2:=data[2,-1]:
data12:=<data1[1..115]; data2; data1[116..257]>:
display(seq(plot(data12[1..n],opt),n=1..257+231), insequence);

 

data := ["curve", [-2. .. 2., -1.70638140024685558 .. 1.56080179572067523], Matrix(257, 2, {(1, 1) = -2.0, (1, 2) = 1.5608017957206752, (2, 1) = -1.9660640560142397, (2, 2) = 1.537635057239961, (3, 1) = -1.96, (3, 2) = 1.5328554865454878, (4, 1) = -1.9407392237181897, (4, 2) = 1.5189159370739755, (5, 1) = -1.92, (5, 2) = 1.5039131407876112, (6, 1) = -1.9122252482478246, (6, 2) = 1.4987599467208406, (7, 1) = -1.88, (7, 2) = 1.4740288968812538, (8, 1) = -1.8598423306197251, (8, 2) = 1.45988483620172, (9, 1) = -1.84, (9, 2) = 1.4437215185644865, (10, 1) = -1.8094367786721437, (10, 2) = 1.4210097256826, (11, 1) = -1.8, (11, 2) = 1.4128476133935193, (12, 1) = -1.76098273203451, (12, 2) = 1.3821346151634795, (13, 1) = -1.76, (13, 2) = 1.381231990869088, (14, 1) = -1.753462139963708, (14, 2) = 1.3757806143767444, (15, 1) = -1.72, (15, 2) = 1.347890406633442, (16, 1) = -1.7138616792316663, (16, 2) = 1.343259504644359, (17, 1) = -1.68, (17, 2) = 1.3133696470241003, (18, 1) = -1.6687217751417596, (18, 2) = 1.3043843941252384, (19, 1) = -1.64, (19, 2) = 1.2775520319317626, (20, 1) = -1.625673807826824, (20, 2) = 1.265509283606118, (21, 1) = -1.6, (21, 2) = 1.2401530205347342, (22, 1) = -1.5847377983800817, (22, 2) = 1.2266341730869974, (23, 1) = -1.56, (23, 2) = 1.200848940983167, (24, 1) = -1.5459485372411552, (24, 2) = 1.1877590625678769, (25, 1) = -1.52, (25, 2) = 1.1592815321976953, (26, 1) = -1.5093590858526778, (26, 2) = 1.1488839520487568, (27, 1) = -1.48, (27, 2) = 1.1150690779074135, (28, 1) = -1.4750453387897324, (28, 2) = 1.1100088415296363, (29, 1) = -1.4542682325784955, (29, 2) = 1.0850007089705542, (30, 1) = -1.4427309336094418, (30, 2) = 1.0711337310105158, (31, 1) = -1.44, (31, 2) = 1.0672743647638947, (32, 1) = -1.4119556558322721, (32, 2) = 1.0322586204913953, (33, 1) = -1.4, (33, 2) = 1.0147459794111668, (34, 1) = -1.3834158233349232, (34, 2) = .9933835099722748, (35, 1) = -1.36, (35, 2) = .9581168441917892, (36, 1) = -1.357261548973772, (36, 2) = .9545083994531542, (37, 1) = -1.352523623693589, (37, 2) = .9472422755732975, (38, 1) = -1.331808942682061, (38, 2) = .9156332889340337, (39, 1) = -1.32, (39, 2) = .8948285347077485, (40, 1) = -1.3082260204822242, (40, 2) = .8767581784149137, (41, 1) = -1.2936906961495054, (41, 2) = .851188751043186, (42, 1) = -1.2860661903102808, (42, 2) = .8378830678957931, (43, 1) = -1.28, (43, 2) = .8258793257046575, (44, 1) = -1.2643432730058457, (44, 2) = .7990079573766726, (45, 1) = -1.248431221937656, (45, 2) = .7683269639734924, (46, 1) = -1.2441342988305302, (46, 2) = .7601328468575521, (47, 1) = -1.24, (47, 2) = .7513826332673788, (48, 1) = -1.2235172573992774, (48, 2) = .7212577363384316, (49, 1) = -1.2080698405455357, (49, 2) = .6902255243962956, (50, 1) = -1.2041119819552997, (50, 2) = .6823826258193111, (51, 1) = -1.2, (51, 2) = .67352975046391, (52, 1) = -1.1837819179192415, (52, 2) = .6435075153001906, (53, 1) = -1.1683659195735205, (53, 2) = .6127630559814374, (54, 1) = -1.1642255855103973, (54, 2) = .6046324047810705, (55, 1) = -1.16, (55, 2) = .5957767133314007, (56, 1) = -1.143272921561723, (56, 2) = .56575729426195, (57, 1) = -1.1258254669844185, (57, 2) = .5325438255639484, (58, 1) = -1.1228029261202637, (58, 2) = .5268821837428295, (59, 1) = -1.12, (59, 2) = .5213499696564032, (60, 1) = -1.100201309604214, (60, 2) = .48800707322370895, (61, 1) = -1.08, (61, 2) = .45228576503715273, (62, 1) = -1.0778963632808125, (62, 2) = .44913196270458844, (63, 1) = -1.074480366362546, (63, 2) = .44376755351256125, (64, 1) = -1.0528397913359078, (64, 2) = .4102568521854679, (65, 1) = -1.04, (65, 2) = .3894939466243382, (66, 1) = -1.0267584599063069, (66, 2) = .3713817416663474, (67, 1) = -1.0, (67, 2) = .33318051018829137, (68, 1) = -.9994387376380839, (68, 2) = .33250663114722734, (69, 1) = -.9975010155015024, (69, 2) = .3300779236831108, (70, 1) = -.9681595746356814, (70, 2) = .2936315206281068, (71, 1) = -.96, (71, 2) = .28306136584737884, (72, 1) = -.9340361658472693, (72, 2) = .2547564101089863, (73, 1) = -.92, (73, 2) = .23877953666160212, (74, 1) = -.8960148047100205, (74, 2) = .2158812995898658, (75, 1) = -.88, (75, 2) = .19989273207119862, (76, 1) = -.8526186408882235, (76, 2) = .17700618907074528, (77, 1) = -.84, (77, 2) = .16595546554269114, (78, 1) = -.802074496710025, (78, 2) = .13813107855162476, (79, 1) = -.8, (79, 2) = .1365329483557478, (80, 1) = -.7950643409078066, (80, 2) = .13333422123428176, (81, 1) = -.76, (81, 2) = .11120709513178723, (82, 1) = -.7373634038441406, (82, 2) = 0.9925596803250425e-1, (83, 1) = -.72, (83, 2) = 0.8957948666131565e-1, (84, 1) = -.7006100437222798, (84, 2) = 0.8041130070107216e-1, (85, 1) = -.68, (85, 2) = 0.7127123926797664e-1, (86, 1) = -.6511226540720738, (86, 2) = 0.6038085751338418e-1, (87, 1) = -.64, (87, 2) = 0.5592405198389689e-1, (88, 1) = -.6328102028220955, (88, 2) = 0.5339325351585669e-1, (89, 1) = -.6, (89, 2) = 0.4319999909229342e-1, (90, 1) = -.5751946069900615, (90, 2) = 0.3627304764509473e-1, (91, 1) = -.56, (91, 2) = 0.32781652827231805e-1, (92, 1) = -.523518178334341, (92, 2) = 0.24924986283600757e-1, (93, 1) = -.52, (93, 2) = 0.24372053193770355e-1, (94, 1) = -.502041186180679, (94, 2) = 0.21505746994263664e-1, (95, 1) = -.48, (95, 2) = 0.17694719998699836e-1, (96, 1) = -.47511178978439206, (96, 2) = 0.16755004184952373e-1, (97, 1) = -.44, (97, 2) = 0.12493653332178958e-1, (98, 1) = -.42892863182938723, (98, 2) = 0.10745730463502565e-1, (99, 1) = -.4, (99, 2) = 0.8533333332284634e-2, (100, 1) = -.3845767047410973, (100, 2) = 0.6516189300291919e-2, (101, 1) = -.36, (101, 2) = 0.5598719999016822e-2, (102, 1) = -.34153209999227924, (102, 2) = 0.35572056478583997e-2, (103, 1) = -.32, (103, 2) = 0.3495253332374052e-2, (104, 1) = -.29941742162404744, (104, 2) = 0.1501996765923273e-2, (105, 1) = -.28, (105, 2) = 0.2048853332352378e-2, (106, 1) = -.2579480624227013, (106, 2) = 0.7395923230780338e-4, (107, 1) = -.24, (107, 2) = 0.11059199989432913e-2, (108, 1) = -.21689631465579817, (108, 2) = -0.9482110346070976e-3, (109, 1) = -.2, (109, 2) = 0.5333333321292256e-3, (110, 1) = -.1760607107060514, (110, 2) = -0.17603159320226328e-2, (111, 1) = -.16, (111, 2) = 0.21845333187214648e-3, (112, 1) = -.13522295659329342, (112, 2) = -0.2574510525055449e-2, (113, 1) = -.12, (113, 2) = 0.6911999807954206e-4, (114, 1) = -0.9403611919942424e-1, (114, 2) = -0.37279713964276755e-2, (115, 1) = -0.8e-1, (115, 2) = 0.13653330468132649e-4, (116, 1) = -0.5151214536331036e-1, (116, 2) = -0.6180965442085072e-2, (117, 1) = -0.4e-1, (117, 2) = 0.8533276100765319e-6, (118, 1) = -0.7393137407817655e-11, (118, 2) = -0.17369363517671627e-1, (119, 1) = .0, (119, 2) = -0.10250177145585194e-1, (120, 1) = 0.9152940627643602e-11, (120, 2) = -0.1736936352485685e-1, (121, 1) = 0.28266988749692245e-11, (121, 2) = -0.17369363527604056e-1, (122, 1) = 0.10552401716878989e-7, (122, 2) = -0.56244474043977366e-1, (123, 1) = 0.6239740617530742e-8, (123, 2) = -0.5624448010824252e-1, (124, 1) = 0.24688489416835327e-6, (124, 2) = -0.9511958456309788e-1, (125, 1) = 0.17526574099147042e-6, (125, 2) = -0.9511975489997417e-1, (126, 1) = 0.19293081909754763e-5, (126, 2) = -.1339946950822184, (127, 1) = 0.14957225746714543e-5, (127, 2) = -.1339961487417283, (128, 1) = 0.8895932177281907e-5, (128, 2) = -.1728698056013387, (129, 1) = 0.7266837047446018e-5, (129, 2) = -.17287686807867228, (130, 1) = 0.30043805849921945e-4, (130, 2) = -.2117449161204592, (131, 1) = 0.2541991452907233e-4, (131, 2) = -.2117696211701268, (132, 1) = 0.8259844995817999e-4, (132, 2) = -.2506200266395797, (133, 1) = 0.7173030514968205e-4, (133, 2) = -.25068973972808634, (134, 1) = 0.1962117914979933e-3, (134, 2) = -.2894951371587, (135, 1) = 0.17404835835773902e-3, (135, 2) = -.28966429088787077, (136, 1) = 0.41788850005981094e-3, (136, 2) = -.3283702476778205, (137, 1) = 0.3778475540232584e-3, (137, 2) = -.3287374693133714, (138, 1) = 0.8177407260297853e-3, (138, 2) = -.36724535819694104, (139, 1) = 0.7537113743273638e-3, (139, 2) = -.36797787352135336, (140, 1) = 0.14955712406446331e-2, (140, 2) = -.40612046871606156, (141, 1) = 0.14079792962529948e-2, (141, 2) = -.4074888524848233, (142, 1) = 0.2588284977899922e-2, (142, 2) = -.44499557923518185, (143, 1) = 0.24988971475248434e-2, (143, 2) = -.44742420180483045, (144, 1) = 0.4278128978311911e-2, (144, 2) = -.48387068975430236, (145, 1) = 0.4262828139857411e-2, (145, 2) = -.4880136376308267, (146, 1) = 0.6801760734532865e-2, (146, 2) = -.5227458002734229, (147, 1) = 0.7059721773361325e-2, (147, 2) = -.5296069868777644, (148, 1) = 0.104601449388199e-1, (148, 2) = -.5616209107925432, (149, 1) = 0.11457587342043786e-1, (149, 2) = -.5727562851476539, (150, 1) = 0.15629278632357448e-1, (150, 2) = -.6004960213116637, (151, 1) = 0.1840207538429938e-1, (151, 2) = -.6183805891703094, (152, 1) = 0.22771744756432996e-1, (152, 2) = -.6393711318307842, (153, 1) = 0.29591217668941294e-1, (153, 2) = -.6681301782626703, (154, 1) = 0.32449094105466564e-1, (154, 2) = -.6782462423499047, (155, 1) = 0.4e-1, (155, 2) = -.7004982010314704, (156, 1) = 0.4533486523659856e-1, (156, 2) = -.717121352869025, (157, 1) = 0.48865811903737635e-1, (157, 2) = -.7257378383090134, (158, 1) = 0.6222882272819009e-1, (158, 2) = -.7559964633881455, (159, 1) = 0.8e-1, (159, 2) = -.7873237889824531, (160, 1) = 0.8407147735783974e-1, (160, 2) = -.794871573907266, (161, 1) = 0.9302075546240601e-1, (161, 2) = -.8075261565983529, (162, 1) = .11195709279581617, (162, 2) = -.8337466844263863, (163, 1) = .12, (163, 2) = -.8423201725159065, (164, 1) = .14714511427182947, (164, 2) = -.8726217949455068, (165, 1) = .16, (165, 2) = -.8836427393581701, (166, 1) = .19106623319433097, (166, 2) = -.9114969054646274, (167, 1) = .2, (167, 2) = -.9176683332647253, (168, 1) = .22468280843107297, (168, 2) = -.9354855781066334, (169, 1) = .24, (169, 2) = -.9464227822334205, (170, 1) = .24528477729825404, (170, 2) = -.9503720159837479, (171, 1) = .28, (171, 2) = -.9703453785152176, (172, 1) = .3113854529800633, (172, 2) = -.9892471265028682, (173, 1) = .32, (173, 2) = -.9932800991411226, (174, 1) = .3283765359695273, (174, 2) = -.9973880955424372, (175, 1) = .36, (175, 2) = -1.0126070712146231, (176, 1) = .39068034039380495, (176, 2) = -1.0281222370219887, (177, 1) = .4, (177, 2) = -1.0318068690600755, (178, 1) = .40663085582836356, (178, 2) = -1.0345666183510884, (179, 1) = .44, (179, 2) = -1.048174782073521, (180, 1) = .47708053370669606, (180, 2) = -1.0641599831708832, (181, 1) = .48, (181, 2) = -1.065394341180254, (182, 1) = .4835697931444164, (182, 2) = -1.0669973475411092, (183, 1) = .52, (183, 2) = -1.0798397321250879, (184, 1) = .5415352254794232, (184, 2) = -1.087926954305278, (185, 1) = .56, (185, 2) = -1.094726962404693, (186, 1) = .5882586552123794, (186, 2) = -1.1058724580602295, (187, 1) = .6, (187, 2) = -1.109540338092825, (188, 1) = .6057654479255884, (188, 2) = -1.1114757686927166, (189, 1) = .64, (189, 2) = -1.1227559165178804, (190, 1) = .6674148621800018, (190, 2) = -1.1325163529880802, (191, 1) = .68, (191, 2) = -1.13692074232505, (192, 1) = .7005865207588209, (192, 2) = -1.14474756857935, (193, 1) = .72, (193, 2) = -1.1506341021836872, (194, 1) = .729176559206252, (194, 2) = -1.1536660624125576, (195, 1) = .76, (195, 2) = -1.1636938208479914, (196, 1) = .7907727688794056, (196, 2) = -1.174654938358506, (197, 1) = .8, (197, 2) = -1.177896342740452, (198, 1) = .8148242297452915, (198, 2) = -1.1836226790984705, (199, 1) = .84, (199, 2) = -1.191437131295202, (200, 1) = .8524055975520314, (200, 2) = -1.1956794034957443, (201, 1) = .88, (201, 2) = -1.2050036322014357, (202, 1) = .9157310556168721, (202, 2) = -1.218348897500239, (203, 1) = .92, (203, 2) = -1.2199267370804074, (204, 1) = .9262764289647889, (204, 2) = -1.222497789617591, (205, 1) = .96, (205, 2) = -1.233683417582458, (206, 1) = .9785172438640943, (206, 2) = -1.2404942871607454, (207, 1) = 1.0, (207, 2) = -1.2483273596456554, (208, 1) = 1.0322112930434906, (208, 2) = -1.2613729001367116, (209, 1) = 1.04, (209, 2) = -1.2639488153585408, (210, 1) = 1.044260604277254, (210, 2) = -1.2655136866906238, (211, 1) = 1.08, (211, 2) = -1.2785467485760127, (212, 1) = 1.110310880683925, (212, 2) = -1.290831371049698, (213, 1) = 1.12, (213, 2) = -1.294734363802857, (214, 1) = 1.1323119065896292, (214, 2) = -1.3002480106558318, (215, 1) = 1.16, (215, 2) = -1.3104450105583172, (216, 1) = 1.1781163382286615, (216, 2) = -1.3178548769268565, (217, 1) = 1.2, (217, 2) = -1.3267614047002712, (218, 1) = 1.2273563967396743, (218, 2) = -1.3391231211749524, (219, 1) = 1.24, (219, 2) = -1.343856183746842, (220, 1) = 1.2485003395080798, (220, 2) = -1.3473844121206184, (221, 1) = 1.28, (221, 2) = -1.3604066145231137, (222, 1) = 1.3184059117598363, (222, 2) = -1.3779982316940727, (223, 1) = 1.32, (223, 2) = -1.3786072058293786, (224, 1) = 1.321108709561705, (224, 2) = -1.3790757618626948, (225, 1) = 1.36, (225, 2) = -1.395458468190525, (226, 1) = 1.3944822106993007, (226, 2) = -1.4115107254910455, (227, 1) = 1.4, (227, 2) = -1.4140723948186662, (228, 1) = 1.4054657151488947, (228, 2) = -1.4168733422131932, (229, 1) = 1.44, (229, 2) = -1.4317193763411282, (230, 1) = 1.4698014480017874, (230, 2) = -1.445836706830676, (231, 1) = 1.48, (231, 2) = -1.4506572107980191, (232, 1) = 1.489787764531519, (232, 2) = -1.4557484527323137, (233, 1) = 1.52, (233, 2) = -1.4690086944071754, (234, 1) = 1.5470781401050524, (234, 2) = -1.4820650949632173, (235, 1) = 1.56, (235, 2) = -1.4882844462646128, (236, 1) = 1.572005316222629, (236, 2) = -1.4946235632514342, (237, 1) = 1.6, (237, 2) = -1.5071635640832501, (238, 1) = 1.6260550342204492, (238, 2) = -1.51994587162392, (239, 1) = 1.64, (239, 2) = -1.5267769815686103, (240, 1) = 1.6525464721089986, (240, 2) = -1.5334986737705547, (241, 1) = 1.68, (241, 2) = -1.5460385692025809, (242, 1) = 1.7065028131276487, (242, 2) = -1.5592561685056783, (243, 1) = 1.72, (243, 2) = -1.5659793129956716, (244, 1) = 1.7317739039734703, (244, 2) = -1.572373784289675, (245, 1) = 1.76, (245, 2) = -1.5855047726854195, (246, 1) = 1.7882109027769322, (246, 2) = -1.59979133337211, (247, 1) = 1.8, (247, 2) = -1.60575550072035, (248, 1) = 1.80998861759019, (248, 2) = -1.6112488948087955, (249, 1) = 1.84, (249, 2) = -1.6254484613306523, (250, 1) = 1.8709822451459517, (250, 2) = -1.6413598499132793, (251, 1) = 1.88, (251, 2) = -1.6459871169518323, (252, 1) = 1.8874374101085565, (252, 2) = -1.6501240053279158, (253, 1) = 1.92, (253, 2) = -1.6657697856527272, (254, 1) = 1.9546306438165593, (254, 2) = -1.6837807579708417, (255, 1) = 1.96, (255, 2) = -1.6865712979596124, (256, 1) = 1.9643212613511274, (256, 2) = -1.6889991158470363, (257, 1) = 2.0, (257, 2) = -1.7063814002468556}, datatype = float[8], order = C_order)], ["curve", [3.29779092922422062*10^(-11) .. 1.23219705364221643, 8.53339056588672817*10^(-7) .. 1.13784600928479618], Matrix(231, 2, {(1, 1) = 0.4e-1, (1, 2) = 0.8533390565886728e-6, (2, 1) = 0.5528668152724739e-1, (2, 2) = 0.6648961148205159e-2, (3, 1) = 0.8e-1, (3, 2) = 0.13653336198533178e-4, (4, 1) = 0.9804170016286295e-1, (4, 2) = 0.3971419799660361e-2, (5, 1) = .12, (5, 2) = 0.6912000192045228e-4, (6, 1) = .13910272761879985, (6, 2) = 0.294023080932616e-2, (7, 1) = .16, (7, 2) = 0.21845333479451716e-3, (8, 1) = .1795002584194989, (8, 2) = 0.2553879464027912e-2, (9, 1) = .2, (9, 2) = 0.5333333345374433e-3, (10, 1) = .21944466847211347, (10, 2) = 0.2607906097736942e-2, (11, 1) = .24, (11, 2) = 0.11059200010567049e-2, (12, 1) = .25895260013158083, (12, 2) = 0.3086136375766102e-2, (13, 1) = .28, (13, 2) = 0.20488533343142896e-2, (14, 1) = .29797313495208877, (14, 2) = 0.4038056803325743e-2, (15, 1) = .32, (15, 2) = 0.34952533342926095e-2, (16, 1) = .33642933914296375, (16, 2) = 0.5538437620792953e-2, (17, 1) = .36, (17, 2) = 0.5598720000983176e-2, (18, 1) = .3742383617337333, (18, 2) = 0.7667799844011202e-2, (19, 1) = .4, (19, 2) = 0.853333333438204e-2, (20, 1) = .41132424553637753, (20, 2) = 0.1049996457495532e-1, (21, 1) = .44, (21, 2) = 0.12493653334487705e-1, (22, 1) = .4476270546408051, (22, 2) = 0.14093182191846937e-1, (23, 1) = .48, (23, 2) = 0.17694720001300155e-1, (24, 1) = .48310908211887815, (24, 2) = 0.18484099219653372e-1, (25, 1) = .5020411861635891, (25, 2) = 0.21505746994263664e-1, (26, 1) = .5172875501190937, (26, 2) = 0.24141916716708914e-1, (27, 1) = .52, (27, 2) = 0.2437205347289633e-1, (28, 1) = .5499125585237185, (28, 2) = 0.3130950705040385e-1, (29, 1) = .56, (29, 2) = 0.3278165383943493e-1, (30, 1) = .5817132569206032, (30, 2) = 0.39278225950421554e-1, (31, 1) = .6, (31, 2) = 0.4320000090770665e-1, (32, 1) = .612751658017325, (32, 2) = 0.4798780464524572e-1, (33, 1) = .64, (33, 2) = 0.559240546827699e-1, (34, 1) = .6431094340915648, (34, 2) = 0.5735886766434661e-1, (35, 1) = .6511226464651612, (35, 2) = 0.6038085751338418e-1, (36, 1) = .6715698118098565, (36, 2) = 0.6857396995310436e-1, (37, 1) = .68, (37, 2) = 0.7127126739872269e-1, (38, 1) = .6990045774844189, (38, 2) = 0.8078584178060508e-1, (39, 1) = .72, (39, 2) = 0.8957955333875738e-1, (40, 1) = .7261120735449809, (40, 2) = 0.9331577966845125e-1, (41, 1) = .7373632461594335, (41, 2) = 0.9925596803250425e-1, (42, 1) = .7517501839350733, (42, 2) = .10727378081466522, (43, 1) = .76, (43, 2) = .11120741153796578, (44, 1) = .7762770701230621, (44, 2) = .12231175605258693, (45, 1) = .8, (45, 2) = .1365337183180539, (46, 1) = .800891953247129, (46, 2) = .13726420902512412, (47, 1) = .8020734588109075, (47, 2) = .13813107855162476, (48, 1) = .8232294772960833, (48, 2) = .15442997664107921, (49, 1) = .84, (49, 2) = .16595877460878689, (50, 1) = .8458328916192436, (50, 2) = .17133733141214158, (51, 1) = .852613707439654, (51, 2) = .17700618907074528, (52, 1) = .867017851512226, (52, 2) = .18962325050169146, (53, 1) = .88, (53, 2) = .19990417642520905, (54, 1) = .887930108472193, (54, 2) = .20817420350773794, (55, 1) = .8959982251157956, (55, 2) = .2158812995898658, (56, 1) = .9078340434524945, (56, 2) = .22770512222374312, (57, 1) = .92, (57, 2) = .23881578615082594, (58, 1) = .9274024295074096, (58, 2) = .24756215347872268, (59, 1) = .9339918498978688, (59, 2) = .2547564101089863, (60, 1) = .9459605189587295, (60, 2) = .2684010695367485, (61, 1) = .96, (61, 2) = .2831697889638833, (62, 1) = .9645277285050479, (62, 2) = .2892311219772491, (63, 1) = .9680587926848883, (63, 2) = .2936315206281068, (64, 1) = .9817237991967729, (64, 2) = .3113937537804841, (65, 1) = .9991903419324721, (65, 2) = .33250663114722734, (66, 1) = .999542558673276, (66, 2) = .33295120820053753, (67, 1) = 1.0, (67, 2) = .3334875387193808, (68, 1) = 1.0154577273117664, (68, 2) = .356358720225864, (69, 1) = 1.0263054731261128, (69, 2) = .3713817416663474, (70, 1) = 1.03180952538374, (70, 2) = .3793418818141263, (71, 1) = 1.04, (71, 2) = .3904192491399243, (72, 1) = 1.0474773456144517, (72, 2) = .40298978625668125, (73, 1) = 1.0520705984598313, (73, 2) = .4102568521854679, (74, 1) = 1.0624086565093642, (74, 2) = .42735348774492493, (75, 1) = 1.0764126961145428, (75, 2) = .44913196270458844, (76, 1) = 1.0777516986745237, (76, 2) = .451317036767293, (77, 1) = 1.08, (77, 2) = .4547640320214163, (78, 1) = 1.091621491665343, (78, 2) = .4767124039015279, (79, 1) = 1.0979521645824486, (79, 2) = .48800707322370895, (80, 1) = 1.1054687308668663, (80, 2) = .5021296905620505, (81, 1) = 1.1194274054824829, (81, 2) = .5268821837428295, (82, 1) = 1.1196224116892073, (82, 2) = .5272491534256496, (83, 1) = 1.12, (83, 2) = .5279189577322693, (84, 1) = 1.132061883885139, (84, 2) = .5540346175343606, (85, 1) = 1.1377779948236464, (85, 2) = .56575729426195, (86, 1) = 1.1445990645666588, (86, 2) = .5807251209386746, (87, 1) = 1.1561177674602001, (87, 2) = .6046324047810705, (88, 1) = 1.15728747962861, (88, 2) = .6072686430116492, (89, 1) = 1.16, (89, 2) = .6130393644151663, (90, 1) = 1.1685944451542758, (90, 2) = .6351547651696158, (91, 1) = 1.1720182213513826, (91, 2) = .6435075153001906, (92, 1) = 1.1794354849914845, (92, 2) = .6634937101433944, (93, 1) = 1.1868485711602554, (93, 2) = .6823826258193111, (94, 1) = 1.1901511905767501, (94, 2) = .6919544646895759, (95, 1) = 1.2, (95, 2) = .7188418049496775, (96, 1) = 1.2005690933926547, (96, 2) = .7207046471250529, (97, 1) = 1.200747711775579, (97, 2) = .7212577363384316, (98, 1) = 1.2089924022194882, (98, 2) = .7513933311046774, (99, 1) = 1.211534240159208, (99, 2) = .7601328468575521, (100, 1) = 1.216733516981503, (100, 2) = .782745024325935, (101, 1) = 1.2207477939021012, (101, 2) = .7990079573766726, (102, 1) = 1.2234990906276844, (102, 2) = .8150448242645416, (103, 1) = 1.2277903901180314, (103, 2) = .8378830678957931, (104, 1) = 1.2288216558175276, (104, 2) = .8487470520336526, (105, 1) = 1.231914985889319, (105, 2) = .8767581784149137, (106, 1) = 1.231882070455498, (106, 2) = .8846478136206373, (107, 1) = 1.2321970536422164, (107, 2) = .9156332889340337, (108, 1) = 1.2310619501026243, (108, 2) = .924319980873681, (109, 1) = 1.227501963074891, (109, 2) = .9545083994531542, (110, 1) = 1.2225886047382204, (110, 2) = .9714301473304985, (111, 1) = 1.2164476636539314, (111, 2) = .9933835099722748, (112, 1) = 1.2, (112, 2) = 1.0269523070801165, (113, 1) = 1.1968095006356547, (113, 2) = 1.0322586204913953, (114, 1) = 1.1724090223354104, (114, 2) = 1.059073678142433, (115, 1) = 1.1614607260711143, (115, 2) = 1.0711337310105158, (116, 1) = 1.16, (116, 2) = 1.0722731086105324, (117, 1) = 1.156800886885541, (117, 2) = 1.0742428779077098, (118, 1) = 1.12, (118, 2) = 1.0970885054078119, (119, 1) = 1.0927210670923824, (119, 2) = 1.1100088415296363, (120, 1) = 1.08, (120, 2) = 1.1145016690124707, (121, 1) = 1.0737439193954272, (121, 2) = 1.1160889871526187, (122, 1) = 1.04, (122, 2) = 1.1249049959174369, (123, 1) = 1.0213702421131177, (123, 2) = 1.128114688949562, (124, 1) = 1.0, (124, 2) = 1.1319740306413681, (125, 1) = .9747724900480682, (125, 2) = 1.1345268974672253, (126, 1) = .96, (126, 2) = 1.1361540130713332, (127, 1) = .9319597203272325, (127, 2) = 1.1372605658112835, (128, 1) = .92, (128, 2) = 1.1378460092847962, (129, 1) = .8917587777064824, (129, 2) = 1.137455857476025, (130, 1) = .88, (130, 2) = 1.1374101680128108, (131, 1) = .8534320159187275, (131, 2) = 1.1358296744653789, (132, 1) = .84, (132, 2) = 1.135168782241518, (133, 1) = .8164909805577931, (133, 2) = 1.132856734754935, (134, 1) = .8, (134, 2) = 1.131409169421205, (135, 1) = .7805980935106541, (135, 2) = 1.1288651230060105, (136, 1) = .76, (136, 2) = 1.1263863826985694, (137, 1) = .7455113998256184, (137, 2) = 1.1240899898557972, (138, 1) = .72, (138, 2) = 1.120325764626022, (139, 1) = .711051887452227, (139, 2) = 1.1187053131354414, (140, 1) = .68, (140, 2) = 1.1134253539230818, (141, 1) = .6770833283938454, (141, 2) = 1.1128434898055672, (142, 1) = .6620349167220663, (142, 2) = 1.1100088415296363, (143, 1) = .6444207891243265, (143, 2) = 1.1057123748848885, (144, 1) = .64, (144, 2) = 1.104699360263677, (145, 1) = .6128018195089421, (145, 2) = 1.0975670378232374, (146, 1) = .6, (146, 2) = 1.0943996235321298, (147, 1) = .581361806444035, (147, 2) = 1.0892477768696383, (148, 1) = .56, (148, 2) = 1.0836594263366455, (149, 1) = .5500505742675701, (149, 2) = 1.0808033566342656, (150, 1) = .52, (150, 2) = 1.0726206176697684, (151, 1) = .5188260743012248, (151, 2) = 1.0722746432925438, (152, 1) = .5147519436763079, (152, 2) = 1.0711337310105158, (153, 1) = .4895027251959482, (153, 2) = 1.061898243704883, (154, 1) = .48, (154, 2) = 1.0586119310046387, (155, 1) = .460380618395989, (155, 2) = 1.0513262611957135, (156, 1) = .44, (156, 2) = 1.0441600900981691, (157, 1) = .4311588851759435, (157, 2) = 1.040851103388831, (158, 1) = .40693985585363684, (158, 2) = 1.0322586204913953, (159, 1) = .40226370069365175, (159, 2) = 1.0300585801251974, (160, 1) = .4, (160, 2) = 1.0290550615454201, (161, 1) = .3752551226757981, (161, 2) = 1.0174325059912854, (162, 1) = .36, (162, 2) = 1.0106738674049787, (163, 1) = .34798174996516995, (163, 2) = 1.0050637799310358, (164, 1) = .3214765874523998, (164, 2) = .9933835099722748, (165, 1) = .3205599404484503, (165, 2) = .9928393163018341, (166, 1) = .32, (166, 2) = .9925278196747336, (167, 1) = .2955759570228421, (167, 2) = .9782455837046733, (168, 1) = .28, (168, 2) = .969702711532461, (169, 1) = .2701747240933576, (169, 2) = .9640573666219436, (170, 1) = .2524697077403972, (170, 2) = .9545083994531542, (171, 1) = .24538168727048992, (171, 2) = .9492780572676632, (172, 1) = .24, (172, 2) = .9455850263231672, (173, 1) = .22228029439955582, (173, 2) = .9328546767736223, (174, 1) = .2, (174, 2) = .9179530273692664, (175, 1) = .19861161728399052, (175, 2) = .9169826272222263, (176, 1) = .1965398873019475, (176, 2) = .9156332889340337, (177, 1) = .17760101591611296, (177, 2) = .8985272529593417, (178, 1) = .16, (178, 2) = .8838847858578841, (179, 1) = .15613067363870786, (179, 2) = .880518690663158, (180, 1) = .15144445697357556, (180, 2) = .8767581784149137, (181, 1) = .13697017179206986, (181, 2) = .8602652458162843, (182, 1) = .12, (182, 2) = .8427002017859868, (183, 1) = .11765657389606378, (183, 2) = .8401605916153909, (184, 1) = .11534870940379771, (184, 2) = .8378830678957931, (185, 1) = .10108201077974491, (185, 2) = .8173939304200963, (186, 1) = 0.8673608546573841e-1, (186, 2) = .7990079573766726, (187, 1) = 0.8451125483170958e-1, (187, 2) = .794623569122607, (188, 1) = 0.8e-1, (188, 2) = .7869256648245326, (189, 1) = 0.703131275221062e-1, (189, 2) = .7695473028116208, (190, 1) = 0.6430159690985951e-1, (190, 2) = .7601328468575521, (191, 1) = 0.5745712443577031e-1, (191, 2) = .7431666558128869, (192, 1) = 0.46927178347976835e-1, (192, 2) = .7212577363384316, (193, 1) = 0.4545720547622919e-1, (193, 2) = .7159539996880826, (194, 1) = 0.4e-1, (194, 2) = .7004315507321014, (195, 1) = 0.3546330485121506e-1, (195, 2) = .6867917389518254, (196, 1) = 0.33654802989824366e-1, (196, 2) = .6823826258193111, (197, 1) = 0.27442873228008802e-1, (197, 2) = .6557115075767848, (198, 1) = 0.23670152754813982e-1, (198, 2) = .6435075153001906, (199, 1) = 0.20604652246946895e-1, (199, 2) = .6234823119674882, (200, 1) = 0.16286504100276277e-1, (200, 2) = .6046324047810705, (201, 1) = 0.1498074978945542e-1, (201, 2) = .5900729471879612, (202, 1) = 0.1093101202954203e-1, (202, 2) = .56575729426195, (203, 1) = 0.10525437108863228e-1, (203, 2) = .5555278559902223, (204, 1) = 0.7131165717047456e-2, (204, 2) = .5268821837428295, (205, 1) = 0.7129435636532549e-2, (205, 2) = .5199532437851004, (206, 1) = 0.450231840575782e-2, (206, 2) = .48800707322370895, (207, 1) = 0.4641830328578904e-2, (207, 2) = .48349578154784617, (208, 1) = 0.27360457064450694e-2, (208, 2) = .44913196270458844, (209, 1) = 0.28936000943734896e-2, (209, 2) = .4463197371179157, (210, 1) = 0.1589332298818813e-2, (210, 2) = .4102568521854679, (211, 1) = 0.1717858501003744e-2, (211, 2) = .40858730370839963, (212, 1) = 0.8745870345108475e-3, (212, 2) = .3713817416663474, (213, 1) = 0.9641527686205409e-3, (213, 2) = .37044470303041144, (214, 1) = 0.45048644191320264e-3, (214, 2) = .33250663114722734, (215, 1) = 0.5063751327611498e-3, (215, 2) = .3320144964159717, (216, 1) = 0.21364663286968622e-3, (216, 2) = .2936315206281068, (217, 1) = 0.2452973226832178e-3, (217, 2) = .29339312161487296, (218, 1) = 0.9112361122095791e-4, (218, 2) = .2547564101089863, (219, 1) = 0.10733487246696427e-3, (219, 2) = .2546520937332436, (220, 1) = 0.33741983203094963e-4, (220, 2) = .2158812995898658, (221, 1) = 0.4112085961969214e-4, (221, 2) = .21584133514080686, (222, 1) = 0.10252069699689948e-4, (222, 2) = .17700618907074528, (223, 1) = 0.1313383954248959e-4, (223, 2) = .17699342458415143, (224, 1) = 0.23154203474584635e-5, (224, 2) = .13813107855162476, (225, 1) = 0.32220638760271926e-5, (225, 2) = .13812794709939277, (226, 1) = 0.31872949536548843e-6, (226, 2) = 0.9925596803250425e-1, (227, 1) = 0.5239338991902542e-6, (227, 2) = 0.9925545883279835e-1, (228, 1) = 0.16154017234981666e-7, (228, 2) = 0.6038085751338418e-1, (229, 1) = 0.4535611931366645e-7, (229, 2) = 0.6038081343278041e-1, (230, 1) = 0.32977909292242206e-10, (230, 2) = 0.21505746994263664e-1, (231, 1) = 0.4e-1, (231, 2) = 0.8533390565886728e-6}, datatype = float[8], order = C_order)]

 

 

 


This animation is quite slow, since it contains many (about 500) points necessary for a good quality graph. By default, 10 frames are played back in one second. If you want faster playback, then open the file, click on the animation plot and in the animation panel replace the number 10 with the number 30, for example. Then the animation will play 3 times faster.

Download animcurve.mw

Try this way:

restart;
alias(X=inttrans:-laplace);

Example of use:

x:=t^2+sin(t): 
inttrans:-laplace(x, s, t);
X(x,s,t);

 

I fixed the error (which Carl indicated), removed one condition (Maple writes that one condition is superfluous), and also reduced  T (if  T>0.2  then Maple returns an error). Now the code is working:

restart;
hBar:= 1:m:= 1:Fu:= 0.2:Fv:= 0.1:
pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;
pdev := diff(v(x, then Maplet),t)+u(x,t)/m*(diff(v(x,t),x))-hBar*(diff(u(x,t),x$2))/(2*m)+v(x,t)*(diff(u(x,t),x))/m = Fv;
ICu:={u(x,0) = 0.1*sin(2*Pi*x)};
ICv:={v(x,0) = 0.2*sin(Pi*x)};
IC := ICu union ICv;
BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t))};
BCv := {v(0,t) = 0.5*sin(2*Pi*t),v(1,t)=-0.5*sin(2*Pi*t)};
BC := BCu union BCv;
pdu := pdsolve({pdeu,pdev},{IC[],BC[]},numeric, time = t,range = 0..1,spacestep = 1/66,timestep = .1);
T := 0.2; p1 := pdu:-plot3d(u,t=0..T,numpoints = 2000,x=0.0..2, shading = zhue,orientation=[-146,54,0], title = print("Figure 1",u(x, t), numeric));

 

Use  Physics:-diff  command for this:

restart;
alias(q = q(x, t), p = p(x, t));                             
H := lambda*p*q+conjugate(lambda)*conjugate(p)*conjugate(q)+(1/2*(p^2+conjugate(q)^2))*(conjugate(p)^2+q^2);                                   
Physics:-diff(H, q);

                   

If you use the kernel  diff command, then the second argument in it should be a name and not a function, for example  should be  diff(sin(x^2), x)  rather than  diff(sin(x^2), x^2)

In my opinion, a more expressive result is obtained by a significantly simpler and shorter code. Why to color the lines if the surface is painted and why to rename the imaginary unit?

restart:
f:= (x,y)-> Re(sqrt(x+I*y)):
A:=plot3d([x, y, f(x,y)],x= -1..1, y= -1..1,grid= [25$2], color=f(x,y), labels= ['x', 'y',  Re(sqrt(x+I*y))], labelfont=[times,14]):
B:=plottools:-reflect(A,[[0,0,0],[1,0,0],[0,1,0]]):
 plots:-display(A,B, axes=frame, orientation=[40,80], lightmodel=light1);

                 

   
 Such coloring emphasizes the mutual opposite of these surfaces to regard to the plane z=0.      

Here is a simple solution without DEtools package. Since you did not specify specific initial conditions, I took arbitrary  x1(0) = x10, x2(0) = x20

restart;
x:=t-><x1(t),x2(t)>;
A:=<1,-2; 4,-5>;
b:=<3,7>;
eq:=diff(x(t),t)=A.x(t)+b;
S:=Equate(op(eq));
ics:=<x10,x20>;
seq(x(0)=~ics);
Sol:=dsolve({S[],seq(x(0)=~ics)}, {seq(x(t))});



Edit. The following extra code converts the Sol (see the code above) into matrix/vector form:

Y1:=coeff(rhs(Sol[1]),exp(-t)); Y2:=coeff(rhs(Sol[1]),exp(-3*t));
Sol1:=simplify(Sol,{Y1=X1,Y2=X2});
Sol2:=(rhs=lhs)~(Sol1);
LinearAlgebra:-GenerateMatrix(Sol2,[X1,X2]);
A1,B:=%[1], x(t)-%[2];
x(t)=A1%.<Y1,Y2>+B;  # The final result

is(convert(Equate(op(value(%))),set)=Sol);  # Check

The final result (although it’s certainly easier to get it out of  Sol  manually):


DE.mw

de:=diff(y(x),x,x) - y(x) = -4*sin(x)^3 + 9*sin(x);
dsolve({de, y(0)=y(2*Pi)}, y(x));  

                  

We see that the solution depends on one arbitrary constant  _C1 .

Here is another way to solve the original problem. We do not add a specific extra condition, but  remove the condition  y(L)=0 , leaving one condition  y(0)=0 . We get a general solution with one arbitrary constant  _С1  (then we re-designate it  as  C ). Then we impose the condition  y(L)=0  and thus get the general solution to the original problem. As an example, we find the first particular solution obtained by Rouben Rostamian, imposing additional conditions  n=10, D(y)(0)=1 :

restart; 	
de := diff(y(x),x,x) + a*y(x) = 0;
Sol:=dsolve({de,y(0)=0}, y(x));
y:=unapply(eval(eval(y(x),Sol),_C1=C), x);
a:=solve(y(L)=0,a, allsolutions);
about(_Z1);
a:=eval(a,_Z1=n);
y(x);
y:=unapply(simplify(%), x) assuming L>0,n::posint; # This is the general solution to de
eval(de,x=0), eval(de,x=L);  # Check

n:=10: # Example of  Rouben Rostamian
C0:=solve(D(y)(0)=1, C);
eval(y(x),C=C0);

  

 
df.mw

If you mean polynomial interpolation, then use  CurveFitting:-PolynomialInterpolation  command. Before this, replace each curly brace with a square one, because in Maple braces creates a set, and we need lists.

data := [[13, -2], [12, -1], [11, 0.0], [10, 1], [9, 2], [8, 3], [7, 
   4], [6, 5], [5, 6], [4, 7], [3, 8], [2, 9], [1, 10]]:
CurveFitting:-PolynomialInterpolation(data, x);

Output:                         11.-1.*x


We see that all these points lie on one straight line  y = 11 - x . Visualization confirms the result:

with(plots):
display(pointplot(data), plot(11-x, x=1..13), view=[0..13,-2..10]);

                      

 

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