I have a question that can majorly help in my physics.
Problem
Say, we have a linear PDE
\begin{equation} \hat{D}~F(x,y)=0, \end{equation}
with $\hat{D}$ being a (second order) differential operator, containing, in general, non-constant coefficients, $\partial_x$, $\partial_y$, $\partial_x^2$, $\partial_y^2$ and $\partial_{xy}$.
Suppose, I know a solution $F(x,y)$ - surface, and I want to cut it with a curve $y=f(x)$, so that I get a function $\psi(x)=F(x,f(x))$.
Question
Is there any general way, to transform the $\hat{D}$ 2D operator to $\hat{D'}$ 1D operator, containing $x$, $\partial_x$ and $\partial_x^2$, so that
\begin{equation} \hat{D'}~\psi(x)=0. \end{equation}
Would much appreciate any close answer or even a probable reference where could I read about this.