I have two problems:
Let $$f : \mathbb{R}^2 \to \mathbb{R}, f \in C^2$$.
Problem 1
Find all functions such that $$\frac{\partial^2f}{\partial x \partial y} = 0$$
Problem 2
Find all functions such that $$\frac{\partial^2f}{\partial^2 x} = \frac{\partial^2f}{\partial ^2y}$$
Attempt to 1
We have by integration
$\frac{\partial f}{\partial x} = c(x)$, where $c(x)$ is a function of $x$ only. Then $f(x, y) = \int c(x) + h(y)$, where $h(y)$ is a function of $y$ only. Since $c(x)$ is arbitrary, we can set $f(x, y) = c_2(x) + h(y)$ with the condition that $c_2(x)$ is derivable. Is it enough?
Attempt to 2
Here I'm quite stuck. I know there are some class of functions that works (like $f(x, y) = k_1(x^2 + y^2) + k_2x + k_3 y + k_4$)) but how to find them all?
The proof for Problem 1 seems fine to me.
Moreover, I believe Problem 2 is a PDE, and you cannot solve it trivially. It is a hyperbolic PDE, exactly the canonical form of the wave equation: given $u(t,x)$, $u$ is such that $$ u_{tt}-v^2u_{xx}=0, $$ $$ u(x,0)=\phi_0(t), $$ $$ u_t(x,0)=\phi_1(t). $$ The general solution is given by D'Alembert, who said that: $$ u(x,t)=\frac{1}{2}(\phi_0(x-vt)+\phi_0(x+vt))+\frac{1}{2v}\int_{x-vt}^{x+vt}\phi_1(s)ds. $$ In your case $v^2=1$. Thus: $$ u(x,y)=\frac{1}{2}(\phi_0(x-y)+\phi_0(x+y))+\frac{1}{2}\int_{x-y}^{x+y}\phi_1(s)ds $$