omyojj
Sep30-11, 02:04 PM
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?
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?