Sturm liouville Boundary Value Problem

Question

Find the complete set of eigenvalues $\lambda$ and their corresponding eigenfunctions of the singular Sturm-Liouville Boundary Value Problem.

$[(1-x^2)u'(x)]' + \lambda u(x) = 0$ on the interval of $ 0<x<1$

$u(0) = 0$

$u(1) < \infty$

$u'(1) < \infty$

I have tried numerous Sturm Liouville Boundary Value problems, but never done problems involving boundary conditions where there are inequalities and infinities, I have tried so far the following:

$$[(1-x^2)u'(x)]' + \lambda u(x) = 0$$

$p(x) = (1-x^2)$, $\space$ $q(x) = 0$ , $\space$ $r(x) = x$;

If $\lambda = 0$

we have $[(1-x^2)u'(x)]' = 0$;

Integrate both sides respect to x which yields,

$(1-x^2)u'(x) = C_1$

$u'(x) = \frac{C_1}{(1-x^2)}$

$u(x) = C_1 ln(1-x^2) + C_2$

The condition that implies $u(0) = 0$ implies $C_1 = 0$, $C_2 = 0$ Hnece $\lambda = 0$ is not an eigenvalue.

From here I am not sure how to proceed to solve the question itself where inequalities are involved with infinites. Any help to this question will be appreciated.

Answer 1

This is a boundary problem for Legendre polynomials (pl. see https://en.wikipedia.org/wiki/Legendre_polynomials) Therefore the eigenvalues have the form of $\lambda=n(n+1)$ where $n$ is arbitrary nonegative integer n.