Can I seek some help on transforming the following heat equation $$0 = \frac{\partial X(x,y)}{\partial x} + \frac{X^2(x,y)}{2k} + \frac{\sigma^2}{4 x^2} \frac{\partial^2 X(x,y)}{\partial y^2}$$
into a homogenous heat equation? The main problem I am stuck with is getting rid of the $\frac{X^2(x,y)}{2k}$ term.
I would know how to proceed from a heat equation with varying diffusion coefficient of the form $$0 = \frac{\partial U(s,y)}{\partial s} + \frac{\sigma^2}{4s^2} \frac{\partial^2 U(s,y)}{\partial y^2}$$
Thanks. The nature of this PDE arises from a project I’ve been working on.