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Use integration to find a solution involving one or more arbitrary 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? I understand if the arbitrary function was something like $f(x,z)=x^z$ then it couldn't be expressed in the form $f(x)+g(z)$. But we don't know what form the arbitrary function will be in, so the more general form $f(x,z)$ will serve better. Is that right?

AntonySC
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Your solution is even broader than two functions of one variable. Indeed, for any two functions $f$ and $g$, you have a function $F(x,z)=f(x)+g(z)$ and one solution $u$ that fits this function $F$.

On top of that, you also have functions $u$ that solve the equation and for which $F(x,z)$ cannot be expressed as $f(x)+g(z)$, for example the function $F(x,z)=z^x$ or $F(x,z)=xz$ or anything else you can think of.

EDIT: Your conclusion is slighly vague. You say that the general form $f(x,z)$ will serve better. That is not true, it's not that it will serve better, it is that the form serves while the form $f(x)+g(z)$ will not serve. You cannot express $f(x,z)=xz$ as $f(x)+g(z)$ for some functions $f$ and $g$.

5xum
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