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finding canonical form for PDE? (Cam Tripos Part II)

This is from Cambridge Part II, 1992, Paper 4, q.22 (on Partial Differential and Integral Equations).

How can the PDE

y^2u_{xx}+2xyu_{xy}+u_{yy}=0

be reduced to canonical form in its hyperbolic region, namely

|x|>1,y\neq0

?

I know the required substitution

(\xi(x,y),\eta(x,y))

should be given by the two solutions of

y^2(f_x)^2+2xyf_xf_y+(f_y)^2=0

, but after substituting

x=\cosh\theta

I got the solution

f(x,y)=C\exp\Big(\frac{y^2}{2}\pm\cosh^{-1}x-\frac{1}{2}e^{\pm2\cosh^{-1}x}\Big)

. Substituting this sort of expression into the PDE looks way too ghastly for a problem that's supposed to take less than half an hour!

Maybe there's a different way of solving

y^2(f_x)^2+2xyf_xf_y+(f_y)^2=0

? It looks as though you should be able to spot a solution, but I can't quite manage! Or can we change this to get a different type of substitution which will still yield the canonical form?

Many thanks for any help with this!

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finding canonical form for PDE? (Cam Tripos Part II)

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