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?