The problem is stated as follows:

---

a) Transform the differential equation

$\displaystyle \frac{\partial^2 f}{\partial x^2} - y^2 \frac{\partial^2 f}{\partial y^2} - y \frac{\partial f}{\partial y}=4y^4,\,$

with the substitution $\displaystyle u=ye^x$ and $\displaystyle v=ye^{-x}$ in the region y>0.

b) Determine all solutions of class $\displaystyle C^2$ (that is, solutions with continuous second (partial) derivatives) for the differential equation in this region. The solutions should be expressed as functions of the variables x and y.

---

My question is about part b), since I don't know how I should get started on that. I'll show some of my work on part a) for some context:

$\displaystyle \begin{cases}

u = ye^x\\

v = ye^{-x}

\end{cases}

\Leftrightarrow

\begin{cases}

x = \ln(\sqrt{\frac{u}{v}})\\

y = \sqrt{uv}

\end{cases}$

(since y>0).

Also, by the looks of the PDE, we have $\displaystyle f(x,y) = f(x(u,v),y(u,v)) = g(u,v)$, so we can use the chain rule:

$\displaystyle \begin{cases}

\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u}\\

\frac{\partial f}{\partial v} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial v}

\end{cases}

$

By finding the partials of x and y with respect to u and v, we get

$\displaystyle \begin{cases}

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

\frac{\partial f}{\partial v} = -\frac{1}{2v} \frac{\partial f}{\partial x} + \frac{\sqrt{u}}{2\sqrt{v}} \frac{\partial f}{\partial y}

\end{cases}

\Leftrightarrow

\begin{cases}

\frac{\partial f}{\partial x} = u \frac{\partial f}{\partial u} - v \frac{\partial f}{\partial v}\\

\frac{\partial f}{\partial y} = \sqrt{\frac{u}{v}} \frac{\partial f}{\partial u} + \sqrt{\frac{v}{u}} \frac{\partial f}{\partial v}

\end{cases}$.

Differentiating again, and substituting the relevant partials back into the original PDE, we eventually, after a few pages of work, get

$\displaystyle \frac{\partial^2f}{\partial u\, \partial v} = -uv$

(assuming I didn't mess up somewhere).

I'm not sure what to make of that, other than how we have a function that depends on u and v, and consequently, on x and y. But, I guess that was kind of the premise for the problem. How do I approach part b)?

I suppose I could integrate with respect to one variable at a time and get $\displaystyle f(u,v) = -\frac{u^2 v^2}{4} + Cu +D$ or $\displaystyle f(u,v) = -\frac{u^2 v^2}{4} + Cv +D$ (C and D being integration constants), but I suspect that won't cover all possible cases.