Change of variables in PDE

Question

I need to use a change of variables in this PDE $f_{xx} - f_{yy} = 0$, using

$s = (x + y)/2$ , $t = (x - y)/2$

I get $f_{ts} = 0$

But I'm asked to deduce that the general solution is of the form f$(x,y) = h(x + y) + g(x - y)$ where h and g are arbitrary functions.

The fact that I'm not really sure how to deduce this could mean that I made an error in my change of variables, or am I missing something?

Answer 1

You just need to go a bit further. The equation $f_{st}=0$ tells you that $f_t$ is constant with respect to $s$, so it must be a function of $t$:

$$f_{t}=G(t).$$

Let the integral of $G$ be $g$. Then this equation tells us that

$$f=g(t)+h(s),$$

since the constant of integration must be constant with respect to $t$. Can you finish?