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The differential equation is question that models an elastic string is:

[tex]\frac {\partial^2 u}{\partial t^2} - \frac{\partial}{\partial x}\left( p(x) \frac{\partial u}{\partial x}\right)= 0[/tex]

Taking [tex]u(x,t) = e^{-i\lambda t} y(x) [/tex]

I simplify above diff.eqn. to: [tex]\lambda^2 y(x) + \frac{\partial}{\partial x}\left( p(x) y'(x)\right)= 0[/tex]

Then taking a WKB ansatz for [tex]y(x,\lambda)[/tex] as follows: [tex]y(x,\lambda) = e^{-i\lambda \xi(x)} \sum_{n=0}^{\infty} \frac{Y_{n}(x) }{\lambda^n}[/tex], I end up with;

[tex]\lambda^2 e^{-i\lambda \xi(x)} \sum_{n=0}^{\infty}\frac{Y_{n}(x) }{\lambda^n}

+ \frac{\partial}{\partial x}\left[p(x)e^{-i\lambda \xi(x)}\left(\sum_{n=0}^{\infty}\frac{Y_n'(x)}{\lambda^n} - i\lambda\xi(x)\sum_{n=0}^{\infty}\frac{Y_n(x)}{\lambda^n}\right)\right][/tex]

The Problem: I wonder if this last equation can be simplified? - I'm trying to obtain an recurrence relation for the eigenfunctions [tex]Y_n(x)[/tex] and their derivatives.

If it can't be simplified, what adjustments / changes are needed for this above method in order to obtain a recurrence relation?

Thanks, dev00790

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# Using WKB(J) method to obtain asymptotics for an elastic string?

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