## Sturm Liouville Problem

Problem: find the eigenvectors and eigenfunctions of the following Sturm-Liouville problem:

$\displaystyle \frac{d}{dx} \left(x^4 \frac{dy}{dx} \right) + \lambda yx^2 = 0, \qquad 1 \leq x \leq 2, \qquad y(1)=y(2) = 0$

Since $\displaystyle x \neq 0$ i can rearrange to

$\displaystyle \displaystyle x^4 \frac{d^2 y}{dx^2} + 4x^3 \frac{dy}{dx} + \lambda yx^2 = 0$

When substituting $\displaystyle y = x^k$ i am having difficulties when simplifying to get k on its own.
Am i doing this correctly?