I am trying to get an analytical solution to this nonlinear second order PDE (I have been told that it's relatively easy to find). The PDE is:
$$u_t - \frac{(\alpha - r)^2 u_x^2 }{2\sigma^2u_{xx}} +rxu_x = 0$$
$$u(x,T) = 0 \hspace{5pt}, \hspace{10pt} x\in [0,\infty]\hspace{5pt}, \hspace{10pt} t\in [0,T]$$
where $\alpha \in \mathbb{R}, \hspace{2pt} \& \hspace{3pt} \sigma^2, r >0$.
Does anyone know how to find this analytical solution? Thank you!