How to solve this differential equation?
\begin{equation} \frac{\partial^2 z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=x-y \end{equation} I have experience with solving differential equations (not partial) so I do not know how to do this. Can you guys also refer me to some resources for learning and practicing this kind of differential equations.
In this specific situation it seems easy to find a solution just considering $z(x,y) = f(x) + g(y)$, $$\frac{\partial^2 z(x,y)}{\partial x^2} - \frac{\partial^2 z(x,y)}{\partial y^2} = \frac{\partial^2 f(x)}{\partial x^2} - \frac{\partial^2 g(y)}{\partial y^2} = x-y $$ and hence it would be enough to solve $$ \frac{\partial^2 f(x)}{\partial x^2} = x \quad \text{and} \quad \frac{\partial^2 g(y)}{\partial y^2} = y. $$ That means there is a trivial solution $f(x) = x^3/6+\gamma x^2+\alpha x+c_1$, $g(y) = y^3/6+\gamma y^2+\beta y+c_2$, and thus $$ z(x,y) = \frac{1}{6}(x^3+y^3)+\gamma(x^2+y^2)+\alpha x+\beta y+C. $$
In the following, I replace the variable $y$ by $t$, to draw attention to the fact that the equation $$\frac{\partial^2 z}{\partial x^2}-\frac{\partial^2 z}{\partial t^2}=x-t$$ is an inhomogeneous wave equation in 1D, with speed $c=1$. By inspection, it has a particular solution $$z(x,t)=\frac{x^3+t^3}{6}$$ Note that if $z_1,z_2$ are solutions, then $w = z_1-z_2$ solves the homogeneous wave equation $$\frac{\partial^2 w}{\partial x^2}-\frac{\partial^2 w}{\partial t^2}=0$$ This has solutions, due to d'Alembert, of the form $$w(x,t) = F(x-t)+G(x+t)$$ where $F$ and $G$ are arbritrary, sufficiently regular functions. You can interpret $F$ and $G$ as plane waves traveling in opposite directions. A general solution to the inhomogeneous equation would therefore have the form $$ z(x,t) = \frac{x^3+t^3}{6} + F(x-t) + G(x+t)$$ for some arbritrary $F$ and $G$.
The following Math SE page suggests some books for self-study. Partial Differential Equations: An Introduction, by W.A. Strauss is supposedly a good first introduction, although I haven't read the book myself.
