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?