Sturm Liouville Problem

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

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

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

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