The fact that the eigenfunctions are orthogonal can be verified easily using the symbolic calculations in MATLAB or MAPLE. But the underlying theory is based on the SturmLiouville problem. You may wish to explore this further by looking at some books.

State what a Sturm-Liouville equation is.

Prove that the eigenfunctions of this equation are orthogonal with the weighting factor.

State how to identify the weighting function directly from the Sturm-Liouville equation in its general form.