Desperate trying to solve a simple Sturm-Liouville equation

1. Sep 30, 2011

omyojj

While dealing with a wave problem,
I encountered the following equation

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

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

$$\frac{dy}{dx} = 0$$

at x = 0 and

$$y=2A$$

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?

Last edited: Sep 30, 2011
2. Oct 1, 2011

jackmell

Here's a numeric approach in Mathematica:

Code (Text):

k = 1.21;
\[Omega] = 3.25;
a = 0.825;

sols =
(First[NDSolve[{(1 - x^2)^2*Derivative[2][y][
x] - 4*x*(1 - x^2)*Derivative[1][y][
x] + (\[Omega]^2 - k^2)*y[x] == 0,
Derivative[1][y][0] == 0, y[a] == 2*a}, y,
x, Method -> {"Shooting",
"StartingInitialConditions" ->
{y[0] == #1, Derivative[1][y][0] ==
0}}]] & ) /@ {-2, 0, 2};

Plot[Evaluate[y[x] /. sols], {x, 0, a},
PlotStyle -> {Black, Blue, Green}]