What is the general solution ( without start terms ) of laplace equation - $u_{ss}=-u_{tt}$

Question

I had a question is Analysis fourier, which was:
let $u(x,y)$ such that $u_{xx}-2u_{xy}+5u_{yy}=0$
when $s=x+y$ and $t=2x$
I solved it, and reached the conclusion of the Laplace equation ( it is true, you can test it, I checked with friends from course, but no need, its a one page and lots of hard algebra ).
We reached:
$$u_{ss}=-u_{tt}$$
Now, how do I solve it general? without start terms, I could not find at google a general solution, there was one that showed 3 solutions, but it was not clear. Please help, thanks.

Answer 1

The general solution: $$u=\psi_0(t-is)+\psi_1(t+is).$$