This question is from my Exterior Differential System course. I have not seen a question that concerned a system so I do not know anyway of attacking it as of yet. Any help appreciated. Thanks !
The first line of attack on any pde is separation of variables. This system is unusual in that it is nonlinear, and yet yields separable solutions. Dym's equation is the only other nonlinear pde I know of that does that. If I were you, I would carry out a separation of variables procedure on this system and see what you get.
Question: what is the characteristic space of a system of pde's?
You might try looking up the term in your textbook's index (or, if you don't have a required text, ask your professor where to find a definition of that term).
As for the solution, like I said, go for a separable solution. That is, assume that
Plug this into your system, and you should be able to get all the x's to one side, and the y's to the other, in both original equations. From that, you can deduce that each of the sides is constant. Use that to convert into 4 ODE's.
From post #8, we will solve
To this we need a BC. From
, then so .
We now introduce an Ampere contact transformation
With this, the derivatives transform as
So now your problem is
which is easily integrated
To determine we need the BC. In our new coordinates it will be
some function (it really doesn't matter - we only need )
and on this boundary
Imposing these two on our solution gives
which eliminating and solving the resulting ODE for gives
. (nb. the one third is to clean things up later)
This in turn gives
Eliminating X and Y gives
Since then we integrate giving
Finally, substitute into both original PDEs and you'll find that you'll need . Also, to get , you'll also need . With this and your final solution is
Whew! It was long but you do have an exact solution to your nonlinear system of PDEs.