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I know the eigenvectors that I get are not orthonormalized, so how do I do this?

Let's say I'm solving a simple Sturm-Liouville problem like [tex]\phi''(x)}+\lambda\sigma(x)\phi(x) = 0[/tex] where [tex]\sigma(x) = 1 - x^{2}[/tex].

The general solution that I have by formulae is

[tex]\phi_{n}(x)\cong\frac{1}{\sigma^{1/4}}sin[\lambda_{n}^{1/2}\int\sigma(s)^{1/2}ds], \lambda_{n}\cong\frac{(n\pi)^{2}}{(\int\sigma(s)^{1/2}ds)^{2})}[/tex]

When I compare the graph of the eigenfunction from my formula to the numerical eigenfunction I got, they are quite similar except it looks like it is missing some weighting function.