Solutions to this PDE?

I have seen a couple of solutions to this PDE -





One is -




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



which gives - <------- why do you think this?





which is where i'm a bit stuck, as substituting



gives a term that i don't know how to simpify.


Any help with both of these would be appreciated

...

Ok first off, i've written the dependent and independent variables the wrong way round.

It should be $\frac{\partial u}{\partial x}=\frac{x}{\sqrt{1+y^{2}}}$

And what's wrong with this latex code?

I have seen a couple of solutions to this PDE -


$\displaystyle \frac{\partial u}{\partial y}=\frac{x}{1+y^2}$


One is -


$\displaystyle u=\ln \left|y+\sqrt{1+y^2}\right|+f(x)$

I see the first one. That's clear. Thanks.

Just worked out the other one. Making stupid errors. Perhaps you could double check?
For the second - let $y=\sinh \theta \Rightarrow dy=\cosh \theta d\theta$

And substitute in, gives $\int du=x\int \frac{1}{\sqrt{1+\sinh ^{2}\theta }}\cosh \theta d\theta$

And from the identity $\cosh \theta =\sqrt{1+\sinh ^{2}\theta }$

So $\int du=x\int \frac{\cosh\theta}{\cosh\theta } d\theta = x\int d\theta= x\sinh^{-1}y+f\left ( x,z \right )$

That should be it. I hope!