I'm quite confused with the following problem. Normally a one-dimensional [Fokker-Planck equation](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation) is written in the following form:

$$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\psi)+\frac{\partial^2}{\partial x^2}(D\psi)$$

While traditional [convection-diffusion equation](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation) without sources has the form:

$$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\psi)+\frac{\partial}{\partial x}(D\frac{\partial \psi}{\partial x})$$

Considering non-constant diffusion $D=D(x,t)$ these equations significantly differ, that looks surprising, because they should interchangeably fit to the same problems (e.g. [here](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation#As_a_stochastic_differential_equation)). Is there any profound reason/physical explanation for such difference?

Or more straightforwardly: both equations are supposed to describe the evolution of $\psi$ with given $F(x,t)$ and $D(x,t)$. Suppose I have my distribution of something $\psi$ and corresponding coefficients, how can I then decide what form of equation I should use?

*P.S. When one writes down a Langevin equation for Brownian motion with non-constant diffusion there appears a so-called noise-induced drift term and the corresponding Fokker-Planck equation then has a form of convection-diffusion equation that I referred earlier.. meaning that "classical" F-P equation is then suitable only for the constant diffusion, which is totally incorrect.. eventually I got lost completely.*