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Black Scholes PDE for non-constant coefficients

I need to derive the Black–Scholes PDE for non-constant coefficients. I suppose we should also use an appropriate transformation such as $y=\ln S$. I have no idea, please help me.
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One step in the derivation of Black-Scholes

One step in the derivation of Black-Scholes Assumptions:(1) ${\displaystyle \frac{\partial F}{\partial t}(t,x)+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}F}{\partial ...
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Black Scholes PDE and its many solutions

I know the general Black-Scholes formula for Option pricing theory (for calls and puts), however I want to know the other solutions to the Black-Scholes PDE and its various boundary conditions. Can ...