I would like to rewrite this general PDE \begin{equation} \alpha\partial_tu+\beta\partial_xu+\gamma\partial_{xx}u+\delta u=\varepsilon \end{equation} in this form $$c\left(x,t,u,\frac{\partial u}{\partial x}\right)\frac{\partial u}{\partial t}=x^{-m}\frac{\partial}{\partial x}\left[x^m f\left(x,t,u,\frac{\partial u}{\partial x}\right)\right]+s\left(x,t,u,\frac{\partial u}{\partial x}\right) $$ where $\alpha, \beta, \gamma, \delta, \varepsilon $ are functions in $x$ or simple coefficients.
For example setting $\alpha=1, \beta=rx, \gamma=\frac{1}{2}\sigma^2 x^2, \delta=-r,\varepsilon=0$ one gets the Black-Scholes equation, while with different settings it is possible to price different financial securities.
My goal is to restate the starting equation to solve it with matlab which adopts the second form.
I read online some guides and tried to understand the examples but how I can switch from one notation to the other is still opaque.
I suppose $s$ to be linked with $\beta$ as there is only the first derivative wrt $x$, $c$ with $\alpha$ as the first derivative wrt $t$ happears.
I can't get exactly how to match each coefficent function in the two formulations and in particular what $m$ and $f$ represent.
Any help or reference will be really appreciate.