- #1
omyojj
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While dealing with a wave problem,
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?
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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