Suppose $u(x,t)$ satisfies the convection-diffusion equation $$u_t+(u(v(x))_x = f(x)u_{xx}$$ Let $u(x,t)=X(x)T(t)$, and find the ODE satisfied by the function X.
I think the first step would be to rewrite the equation to
$XT'+(X'T(v(x)) = f(x)X''T$
(Please correct me if I am wrong)