The Riccati differential equation and a diffusion-type equation

Construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The heat kernel is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of a Riccati differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.