Say I want to solve an Sturm-Lioville problem with non-homogeneous term but homogeneous boundary conditions:
$y''+(1/x)y'+(1/x^2)y=x$
$y'(0)=0$
$y(1)=0$
One method of solution involves finding the eigenfunctions associated with the homogeneous problem:
$\phi''+(1/x)\phi'+(\lambda/x^2)\phi=0$
$\phi'(0)=0$
$\phi(1)=0$
and then expanding the solution of the initial problem in terms of the eigenfunctions.
$y=\sum a_{n}\phi_{n}(x)$
We can plug this series into the original differential equation to find the coefficients of the series. For that to be valid, term-by-term differentiation of the eigenfunction series must be valid. However, every single book in the topic just states that "The eigenfunction series can be differentiated if the eigenfunctions satisfy homogeneous boundary conditions". I'm trying to search for a proof for it, doesn't have to be formal (I'm an engineering grad student), but something more than just a statement without any justification.
