- #1

AntSC

- 65

- 3

[tex]\frac{\partial u}{\partial y}=\frac{x}{\sqrt{1+y^2}}[/tex]

for a function [itex]u(x,y,z)[/itex]

[tex]u(x,y,z)=x\int \frac{dy}{\sqrt{1+y^2}}[/tex]

let [itex]y=\sinh v[/itex]

[tex]u(x,y,z)=x\int \frac{\cosh v\: dv}{\sqrt{1+\sinh ^2v}}[/tex]

[tex]u(x,y,z)=x\sinh ^{-1}y+f(x,z)[/tex]

So here's the question. Why is the solution with an arbirary function [itex]f(x,z)[/itex] and not two arbitrary functions [itex]f(x)+g(z)[/itex]? What's the difference?