Re: Wrong ODE solution in Mathematica 7?

*To*: mathgroup at smc.vnet.net*Subject*: [mg106197] Re: [mg106177] Wrong ODE solution in Mathematica 7?*From*: "David Park" <djmpark at comcast.net>*Date*: Tue, 5 Jan 2010 01:42:36 -0500 (EST)*References*: <7588881.1262603794881.JavaMail.root@n11>

Remember that the constants are arbitrary and so it is possible to obtain different looking forms depending on how you pick the constants. Clear[y] DSolve[y''[x] == -Cos[x]/(1 + Sin[x])^2, y, x][[1, 1]]; y[x_] = y[x] /. % C[1] + x C[2] + (2 Sin[x/2])/(Cos[x/2] + Sin[x/2]) Check the solution: y''[x] == -Cos[x]/(1 + Sin[x])^2 // Simplify True Here is your second solution: y2[x_] = C[1] + C[2] x - 2/(Tan[x/2] + 1); Check it: y2''[x] == -Cos[x]/(1 + Sin[x])^2 // Simplify True Add the constant 2, which could be absorbed into C[1], to the y2 solution and see if it is equal to the y solution: y2[x] + 2 == y[x] // Simplify True How did I figure out how to add 2? By plotting and trying various constants. Start out with zero and you can see that you need to shift the second curve up. Plot[{(2 Sin[x/2])/( Cos[x/2] + Sin[x/2]), -(2/(Tan[x/2] + 1)) + 2}, {x, -2 \[Pi], 2 \[Pi]}] David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: Zsolt [mailto:phyhari at gmail.com] Hi! I tried solve the ODE: DSolve[D[y[x], x, x] == -Cos[x]/(1 + Sin[x])^2, y[x], x] The solution what M7 (and Wolfram Alpha) gives is: y[x] -> C[1] + x C[2] + (2 Sin[x/2])/(Cos[x/2] + Sin[x/2]) I think, it's wrong! (Does anybody know how to check?) Another system gives for the same diff.eq: y(x) = -2/(tan((1/2)*x)+1)+_C1*x+_C2 (similar, but not the same->ctan vs tan...) I found the problem in one of my math books, and the solution there concours with the other system. How can I trust Mathematica, if it makes mistakes in such simple things?? :( Thank you for your answer! :)