I found the following differential equation:
$$\frac{\partial}{\partial t}f\left(x,t\right)=xf\left(x,t\right)+g\left(x\right)\int_{0}^{1}f\left(x',t\right)x'\mathrm{d}x'$$
to solve for $f(x,t)$. Here $g(x)$ is a given function of $x$, and we know the initial condition $f(x,0)$.
Before I try to solve this equation numerically, anyone knows if there is an analytical solution available?