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

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# Solutions to this PDE?

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