a non-linear second-order PDE

From: andreas wagener (108076_at_gmx.net)
Date: 09/24/04 

Date: Fri, 24 Sep 2004 01:00:06 +0000 (UTC)

I consider the following non-linear second-order partial differential
equation with two non-negative variables x and y:

(a+ b x) [ df/dx d^2f/(dx dy) - df/dy d^2f/(dx^2) ] + b y [ df/dx d^2
f/d(y^2)- df/dy d^2f/(dx dy)] =0

where a >0 and b is a real number. Additional conditions are

f(x,0) = (a + b x)^(1-1/b) /(1/b (1-1/b))

and

df(x,0)/dy =0.

for all x.

(For b=1, f(x,0) converges to logarithmic function ln(a+x), but that
should not really matter).

One solution to the PDE is:

f(x,y) = 1/(1/b (1-1/b)) [ (a + b x)^(1- 1/b) + c2 b y^(1-1/b) ]

with constants c1, c2.

My question: Are there other solutions? If so, which? If no, how can
one prove uniqueness?

Best regards,

Andreas