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Re: Wrong ODE solution in Mathematica 7?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106207] Re: [mg106177] Wrong ODE solution in Mathematica 7?
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 5 Jan 2010 01:44:40 -0500 (EST)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <201001041058.FAA21179@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

Mathematica has NOT made any error there!  How would YOU check that a 
purported solution of an ODE is in fact a solution?  I hope you would 
know to do that by simply "plugging in".  (And if you don't know that, 
then you have no business trying to solve differential equations, with 
any CAS or by hand!)

Let's try it:

    y[x_] = C[1] + x*C[2] + (2*Sin[x/2])/(Cos[x/2] + Sin[x/2])
    y''[x] == -Cos[x]/(1 + Sin[x])^2 // Simplify
True

(I presume you trust that Mathematica knows how to take derivatives and 
do simple algebraic and trig simplifications!)

Now what about the solution provided by the "other system"?  Let's plug 
that in and check:

    z[x_] = -2/(Tan[(1/2)*x] + 1) + C1*x + C2
    z''[x] == -Cos[x]/(1 + Sin[x])^2 // Simplify
True

Hmm... two solutions for the same 2nd order ODE. If we specify a set of 
two initial conditions, we hope the solution will be unique (on some 
suitable interval).  Let's use the values of the function and its 
derivative at 0.

    cstRules=First@Solve[{y[0],y'[0]}=={z[0],z'[0]},{C1,C2}]
{C2 -> 2 + C[1], C1 -> C[2]}

    w[x_] = z[x] /. cstRules;
    w[x] == y[x] // Simplify
True

Tada!  I leave it as an exercise to see why the two solutions are just 
different forms of the same function.


Zsolt wrote:
> 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! :)
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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