Solutions to PDE: Understanding and Simplifying the Process

[AntSC]

I have seen a couple of solutions to this PDE -

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

One is -

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

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 y=\sinh \theta giving -

u=x\ln \left | \cosh \theta \right | + f\left ( x \right )

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

Any help with both of these would be appreciated

 

[HallsofIvy]

I have seen a couple of solutions to this PDE -

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

I am confused by your notation. You have x differentiated with respect to u but on the right both x and y. If you really mean what you have, it is not a "partial" differential equation at all. We are thinking of x as a function of u, with y a fixed parameter. Further that is a separable equation: dx/du= x\sqrt{1+ y^2} gives
\frac{dx}{x}= \frac{du}{\sqrt{1+ y^2}}
ln|x|= \frac{u}{\sqrt{1+ y^2}}+ c
x= Ce^{\frac{u}{\sqrt{1+ y^2}}}
where C= e^c.

One is -

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

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 y=\sinh \theta giving -

u=x\ln \left | \cosh \theta \right | + f\left ( x \right )

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

Any help with both of these would be appreciated

Your original equation treats x as a function of u but your answer gives u as a function of x. Which is intended? if the latter, then
\frac{du}{dx}= \frac{\sqrt{1+ y^2}}{x}
\frac{du}{\sqrt{1+ y^2}}= \frac{dx}{x}
\frac{u}{\sqrt{1+ y^2}}= ln|x|+ C
u= (ln|x|+ C)\sqrt{1+y^2}
essentially just solving the previous equation for u.

 

[AntSC]

Sorry, I've written the dependent and independent variables the wrong way round and got the independent variable wrong. It's been a long week!

It should be -

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

for a function u\left ( x,y,z \right )

There will be a constant term in the form f\left ( x,z \right ) in addition to whatever the result of the integration yields (which i can't work out).

 

[HallsofIvy]

Since there is no differentation with respect to x, you can treat x as a parameter:
\frac{du}{dy}= \frac{x}{\sqrt{1+ y^2}}
is separable
du= x\frac{dy}{\sqrt{1+ y^2}}

 

[AntSC]

I've managed to get it sussed. Thanks everyone for the help. Even if it was for me to see that I've written the question down wrong!