Solving the differential equation $\frac{∂^2z}{∂x^2}−\frac{∂^2z}{∂y^2}=x−y$ 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.
 A: 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
$$ z(x,y) = \frac{1}{6}(x^3+y^3)+\gamma(x^2+y^2)+\alpha x+\beta y+C. $$
A: 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.
