Consider an arithmetic Brownian motion $X_t$ which follows $dX_t=\mu dt+\sigma dZ_t$ where $\mu$ and $\sigma$ are constants and $r$ is the discount rate. Assume an asset price $S_t=X_t^2$. I need to find the stochastic differential equation satisfied by the process $S_t$ and the density and distribution functions of $S_t$.
I first find the SDE using Ito's lemma on $S_t$ and get \begin{align*} dS_t &= \left(\frac{\partial S_t}{\partial t}+\mu\frac{\partial S_t}{\partial X_t}+\frac{1}{2}\sigma^2\frac{\partial^2 S_t}{\partial X_t^2}\right)dt+\sigma\frac{\partial S_t}{\partial X_t}dZ_t\\ &= \left(0+2\mu X_t+\sigma^2\right)dt+2\sigma X_tdZ_t\\ &=\left(2\mu\sqrt{S_t}+\sigma^2\right)dt+2\sigma\sqrt{S_t}dZ_t \end{align*}
My question is how to find the distribution of $S_t$ now. I know that $X_t$ is normally distributed in that $X_T-X_0\sim N(\mu T,\sigma^2T)$, but I'm not exactly sure how to pull the distribution of $S_t$ out of the SDE.
I recognize from other classes that a square normal is going to be a generalized $\chi^2$ distribution, however I think that is out of the scope of this class so I think I must be reasoning wrongly about something else.