Question

Find the cononical from of
uxx - x uyy =0 x>0

Answer 1

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General Solution for x2uxx−y2uyy=0

calculus partial-differential-equations

I tried finding the General solution of the PDE: x2uxx−y2uyy=0

I first tried reducing it to canonical form and then I got stuck.

Here was what I did:

I got the characteristic equation to be:

dydx=±yx

Then solving further, I got:

lny=lnx+C1

and

lny=−lnx+C2

So in order to reduce the PDE into its canonical form, I introduced the new functions: ξ,η

Such that:

ξ=lny−lnx

and

η=lny+lnx

Thus the function becomes:

u=[ξ(x,y),η(x,y)]

So I got uxx & uyy in terms of uξ,uη,uξη,uξξ,uηηand slotted it into the PDE and I got this:

uξ−2uξη=0

Is that the right canonical form for the PDE? And if it, how can I solve further to get the General Solution