Find the eigenfunctions and eigenvalues of the equation with boundary conditions and
This is a Sturm-Liouville Problem.
The aux equation is
I have found that for and the solutions are trivial.
For though
we get complex roots
so the ODE solution is
Plugging in the first boundary condition we will get
so
Plugging in the second boundary condition we will get
Now
For a non-trivial solution we assume the which implies
Since zeros of Sin only occur at integral multiples of , to satisfy the boundary condition we must have
for
Then is the eigenvalue, but Im having trouble with the eigenfunction......
Can anyone please help?