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**pickslides** I think of this problem as find the solution $\displaystyle u(x,y)$ such that $\displaystyle u(x,y) = u_{xx}$

As you only have a partial with respect to the same variable then the solution is similar to that of a 2nd order ODE. I.e $\displaystyle y'' = y$ has the solution $\displaystyle y=e^x$ and $\displaystyle y= e^{-x}$

The difference in your problem is that $\displaystyle f(y)$ and $\displaystyle g(y)$ need to be introduced but are constant with respect to $\displaystyle x$