solution to the pde $u_{xx} -u_{yy}=0$

Question

I am looking at solutions for the following PDE

$u_{xx}-u_{yy}=0$ One solution would be $u(x,y)=g(x-y)+h(x+y)$ with the two times differentiable functions $h,g$.

How do I prove that every solutions of the PDE are of that particular form ?

Edit: I try to answer this myself now. I got the following.

Using $s=x+y, t=x-y$ the PDE above is equivalent to $u_{st}=0$

Hence, $u_t=g'(t)$ $\implies u=g(t)+h(s)$ for some functions $g,h$.

I got one follow-up question. Why are those two functions two-times differentiable ?

Answer 1

You can view this as a wave equation with $c=1$. Look up d'Alembert's formula to find the proof you're interested in.