Finding Solutions to Differential Equations with Constant Coefficients

[azzaz]

What is the general method for solving a differential equation of
the form

\begin{equation}
\frac{\partial^{2}z}{\partial x^{2}}+\frac{\partial^{2}z}{\partial y\partial x}=C\end{equation}

where C is a constant.

 

[mfb]

You can define $$w=\frac{\partial z}{\partial x}$$That simplifies your equation. Defining a=x+y and b=x-y and rewriting the equation in terms of those should make it easier again. Looks like there is a lot of freedom in the solution with just this condition. z can be recovered by integration later.