Can an ordinary differential equation contain more than one ordinary derivative to more than one independent variable?

Question

If we have this equation:$$\frac{df}{dx} + \frac{dg}{dy}=f+g$$Is this an ODE? If yes, can we consider it linear?

Answer 1

Suppose that $I$ is an intervall in $\mathbb R$, $f,g:I \to \mathbb R$ are differentiable and that $f'(x)+g'(y)=f(x)+g(y)$ for all $x,y \in I$.

Then we have $f'(x)-f(x)=g(y)-g'(y)$ for all $x,y \in I$.

Thus there is a constant $c$ such that

$f'(x)-f(x)=c$ for all $x \in I$ and $g(y)-g'(y)=c$ for all $y \in I$.

Can you proceed ?