My professor has said that this will be an easy homework exercise. He suggested using change of variable $t = \dfrac{2}{3}x^{3/2}$, and then removing the first derivative term of the form $p(t) \dfrac {dy}{dt}$ by a transformation
\begin{equation} y(t) = w(t)e^{\frac{-1}2\int p(t)dt} \end{equation}
Whenever I try do this I get a very messy bunch of terms that have the exponential in them. Is this problem really that easy to do?
After change of variables I get:
\begin{equation*} y^{(2)} - xy = (3/2)^{2/3}t^{2/3}*{\frac{d^2y}{dt^2}} + \frac{1}2 (3/2)^{-1/3}t^{-1/3}{\frac{dy}{dt}} - (3/2)^{2/3}t^{2/3}y \end{equation*}
Then I try to use \begin{equation} y(t) = w(t)e^{\frac{-1}2\int p(t)dt} \end{equation}
by taking derivatives and substituting into the above equation.. then its just a mess of terms with exponentials. The second derivative of y(t) in that transformation is really ugly.. so I think I'm doing it wrong.