Kind of heat equation

Question

How to solve the following equation or at least in what direction should I think? $$\frac {\partial u}{\partial t}=\frac {\partial^2 u}{\partial x^2}+tu$$ It's almost exactly the heat equation, but this $tu$ makes things bad. I don't know how to deal with it.

Answer 1

Define a new function $v(x,t)$ via $$ u(x,t) \;=\; e^{t^2/2}\, v(x,t)\, , $$ and this reduces to the ordinary heat equation in $v$: $$ \frac{\partial v}{\partial t} \;=\; \frac{\partial^2 v}{\partial x^2} $$

Then just do whatever you would normally do to solve the heat equation.