Cauchy-Euler Boundary Value Problem

Question

Find the eigenvalues λ_n and eigenfunctions y_n(x) for the given boundary-value problem:

x²y’’ + xy’ + λy = 0,   y (1) = 0,   y (e^π ) = 0

Any help would be greatly appreciated!

Answer 1

We use solution in the form y = x^m. We get equation: m² + λ = 0, m = ±√(- λ).

Then if λ < 0 then y = C₁e^{x√(- λ)} + C₂e^{-x√(- λ)} and easy to see that no solution of boundary value problem

If λ = 0 then then y = C₁x⁰ + C₂x⁰lnx = C₁ + C₂lnx.

Then C₁ + C₂ln1 = 0 and C₁ = 0; C₂lne^π = πC₂ = 0 and C₂ = 0. So again y = 0

If λ > 0 then m = ± i√λ and y = C₁cos(lnx√λ) + C₂sin(lnx√λ)

Then we have at x = 1 C₁ = 0 so y = C₂sin(lnx√λ) and x = e^π y = C₂sin(π√λ) = 0; π√λ = πn, n positive integer

and λ = n² is eigen values, n = 1, 2, 3, ...

then igenfunctions is y_n = sin(nlnx) is eigen functions