Separation of variables to solve PDE with variable coefficients

Question

If the coefficients of the PDE $$\partial_t u = a(x,t)\partial_{xx}u(x,t) + b(x,t)\partial_x u(x,t) + c(x,t)u(x,t) + d(x,t)$$ are in some Hölder space, apparently we can solve this via separation of variables. But how can this be? For example $a(x,t)$ could be something horrible so we can't get terms depending on $x$ on one side and terms depending on $t$ on another side. So what is meant by separation of variables in this context? How does it work?

Answer 1

In fact this should be not the business about Hölder space but only the business about the partial derivatives combinations and the coefficients of the terms in the PDE.

$\partial_tu(x,t)=a(x,t)\partial_{xx}u(x,t)+b(x,t)\partial_xu(x,t)+c(x,t)u(x,t)+d(x,t)$ is separable iff we can find $u_p(x,t)$ of the subsititution $u(x,t)=u_c(x,t)+u_p(x,t)$ so that the PDE becomes $\partial_tu_c(x,t)=a(x,t)\partial_{xx}u_c(x,t)+b(x,t)\partial_xu_c(x,t)+c(x,t)u_c(x,t)$ (since separation of variables can only work on homogeneous PDEs but not on inhomogeneous PDEs) and $a(x,t)=a(x)f(t)$ and $b(x,t)=b(x)f(t)$ and $c(x,t)=c(x)f(t)$ .