While dealing with a wave problem,(adsbygoogle = window.adsbygoogle || []).push({});

I encountered the following equation

[tex] \frac{d}{dx}\left[(1-x^2)^2\frac{d}{dx}y\right] - k^2y = -\omega^2y [/tex]

with x ∈ [0,A], (0<A<=1)

where k is a real number.

Thus it has eigenvalue ω^2 and weight unity.

Boundary conditions are

[tex] \frac{dy}{dx} = 0 [/tex]

at x = 0 and

[tex] y=2A[/tex]

at x= A.

I only need to obtain the solution for the ground state (the one with lowest eigenvalue).

for general values of k>0, 0<A<1.

I find from the physical point of view that the solution should look like

y_0=constant for A->1,

and y_0 = cosh(kx) for A -> 0

Can anybody give me a hint on how to solve this equation?

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# Desperate trying to solve a simple Sturm-Liouville equation

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