I have seen a couple of solutions to this PDE -(adsbygoogle = window.adsbygoogle || []).push({});

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

One is -

[itex]u=\ln \left | y+\sqrt{1+y^{2}} \right |+f\left ( x \right )[/itex]

I have no idea how this is arrived at or if it's correct. This is what i want to know.

The solution i've checked out makes the substitution of [itex]y=\sinh \theta [/itex] giving -

[itex]u=x\ln \left | \cosh \theta \right | + f\left ( x \right )[/itex]

which is where i'm a bit stuck as substituting in [itex]\theta =\sinh^{-1} y[/itex] gives a [itex]\cosh \sinh^{-1} y[/itex] term that i don't know how to simpify.

Any help with both of these would be appreciated

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Solutions to this PDE?

**Physics Forums | Science Articles, Homework Help, Discussion**