Finding General solution to pde $u_x^2+u_y^2=0$ $$u_x^2+u_y^2=0$$
I'm having trouble trying to find a general solution to the pde
What I did so far was I rewrote it as $(d_x + d_y)u^2$. So I need to find when $d_x =-d_y$ Im just really stuck on where to go. I found a general solution by trial and error to be $Ae^{k(x-y)}$ where $k$ is an element of the reals.

Seems that you misunderstood the ambiguous notation $u_x^2$ which meaning is usually $(u_x)^2$, not $(u^2)_x$. The sum of two positive reals is zero if and only if both are zero themselves. Therefore, for real $u$, the equation $u_x^2 + u_y^2 = 0$ implies that both partial derivatives vanish:
$$
u_x = 0 \qquad\text{and}\qquad u_y = 0 .
$$
Therefore, $u(x,y) = c_1$ is constant.
