Sorry for the poor lay-out to follow (i'm useless at latex) but I think I'm missing a trick here, I've got the answer but I can't get to it.
The function u(x, y) satisfies the partial differential equation;
6d^2u/dx^2 + y(d^2/dxdy) - y^2 (d^2u/dy^2) + 2(du/dx) = 0.
Using the method of characteristics (told to) I got
p = x - 2ln (y) r =x + 3ln(y) (just use those letters as I don't have the greek letters on my computer that are used.)
Using the chain rule show that this equation transforms to 5(d^2u/d!d?) + du/di = 0.
I've got it down to
d^2u/dx^2 = d^2u/dp^2 + 2(d^2u/drdp) + d^2u/dr^2
d^2u/dy^2 = 9/y^2(d^2u/dr) - 12/y^2(du/drdp) + 4/y^2(d^2u/dp^2) - 3/y^2 (du/dr) + 2/y^2 (du/dp)
d^2u/dxdy = 3/y(d^2u/dr^2) + 1/y(d^2u/dpdr) - 2/y(d^2u/dp)
When substituting these values back into the original equation the terms I want to cancel out do and as its an hyperbolic I want to be left with the du/dxdy terms.
But instead of getting 5 as the co-efficient like the answer, I keep getting 25, I can't for the life of me see where I'm dropping the minuses. Spent about an hour and a half on this question so far and driving me crazy.
I see where I went wrong now, it was all in the first order terms with what I was being cocky about and ignoring .
All I needed to do was pay more attention to the signs there and realised I had to divide it by 5!!
Aye, thats the next part, I've solved it as it gives you the answer in the question but it was driving me crazy as I couldn't see where I was going wrong to get the pde.BTW - this PDE is totally solvable.