Partial Differential/Integration Arbitrary Functions

[AntSC]

Use integration to find a solution involving one or more arbirary functions
$$\frac{\partial u}{\partial y}=\frac{x}{\sqrt{1+y^2}}$$
for a function $u(x,y,z)$
$$u(x,y,z)=x\int \frac{dy}{\sqrt{1+y^2}}$$
let $y=\sinh v$
$$u(x,y,z)=x\int \frac{\cosh v\: dv}{\sqrt{1+\sinh ^2v}}$$
$$u(x,y,z)=x\sinh ^{-1}y+f(x,z)$$
So here's the question. Why is the solution with an arbirary function $f(x,z)$ and not two arbitrary functions $f(x)+g(z)$? What's the difference?

 

[AlephZero]

$f(x,z)$ is more general than $f(x) + g(z)$.

For example, how would you convert a function like $f(x,z) = x^z$ to the form $f(x) + g(z)$?

 

[AntSC]

$f(x,z)$ is more general than $f(x) + g(z)$.

For example, how would you convert a function like $f(x,z) = x^z$ to the form $f(x) + g(z)$?

Good point. But how can i explain that in a mathematical language?