How does one find the eigenvalues and eigenfunctions of
-(x^2y')'=lambda *y
with 1<x<e, and y(1)=y(e)=0
I'm completely lost. Thank you!
This equation is of type Cauchy-Euler
Using the anzant
gives the characteristic equation
From the boundary conditions we can see we need periodic solutions so the roots of the characteristic equation must be complex. Solving the above for r gives
This implies that lambda must be positive
This gives solutions
Using the boundary conditions to solve for lambda gives
This gives
Now with
This gives