# Sturm liouville Boundary Value Problem

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

$$u(0) = 0$$

$$u(1) < \infty$$

$$u'(1) < \infty$$

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.

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.