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

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

Since <br />
x \neq 0<br />
i can rearrange to

<br />
\displaystyle<br />
x^4 \frac{d^2 y}{dx^2} + 4x^3 \frac{dy}{dx} + \lambda yx^2 = 0<br />


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