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I consider PDE for the function $f(x, t)$ of the form:

\begin{equation} \begin{aligned} f_t + Ax^2 f_{xx} + Bx f_{x} + g(x)f + h(x, t) = 0 \\ f(x, T) = 0 \end{aligned} \end{equation}

where $f:\mathbb{R}^{+}\times [0, T] \rightarrow \mathbb{R}$.

This is very general form of PDE, but I wonder what kind of substitution or other method I can use to simplify it and finally solve.

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  • $\begingroup$ The $x$ part looks like a Cauchy-Euler form, so maybe try $x = e^y$? $\endgroup$ – Dylan Mar 6 at 11:04

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