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Q: Combining NDSolve with FindRoot

• To: mathgroup at smc.vnet.net
• Subject: [mg13282] Q: Combining NDSolve with FindRoot
• From: trivedi at yukawa.uchicago.edu (Anil Trivedi)
• Date: Fri, 17 Jul 1998 03:18:32 -0400
• Organization: University of Chicago
• Sender: owner-wri-mathgroup at wolfram.com

I am new to mathematica and would appreciate any help with a conceptual
obstacle I am encountering. 

Trying to learn mathematica, I thought I would verify the  following.
The "harmonic oscillator" equation:

	y''[x] + (2e-x^2) * y[x] =0, 
	y[0]=1, y'[0]=0

has solutions which vanish for large x only if e is one of the
eigenvalues e= 0.5, 2.5, 4.5, 6.5, etc.. How can I generate this
series, or the exact function e[n] = 2n+1/2 where n=0,1,2,..?

Focussing on the first eigenvalue e = 0.5, let us try to (i) solve the
equation with NDSolve,  (ii) evaluate the soln at some large x = L,
(iii) call the resulting function z[e], and  (iv) use FindRoot to solve
z[e]=0, with a good intitial guess  like 0.45. :)

This plan sounds nice but here are the obstacles:

1. NDSolve needs numerical coefficients. How to use it with a parameter
like e?

2. How to use FindRoot with an implit function like z[e] above?  All
examples I have seen are with an explicit function.

3. If I want to specify the accuracy of the answer (say 6 decimal
places), where do I do it? In FindRoot, NDSolve, or both?

3. Assuming I can do this for one eigenvalue, what is the best
"mathematica way" of iterating the procedure to obtain the first N
eigenvalues? (I doubt it is Do loop, but I don't know what it is.)

Thanks,

Anil Trivedi