# Homework Help: PDE i.v.p. using method of characteristics

1. Jun 26, 2012

### sunrah

1. The problem statement, all variables and given/known data
solve x2ux + y2uy = 0 for u(2,y) = y

2. Relevant equations
3. The attempt at a solution

with a = x2 and b = y2

y' = b/a = (y/x)2 this can be solved for y by separation of variables:

$y = \frac{x}{1-xC}$
and
$C = \frac{y-x}{xy}$

now

$u(x,y) = f(C) = f(\frac{y-x}{xy})$

applying initial value conditions

$u(2,y) = f(\frac{y-2}{2y})$

this is where my understanding runs out. how do i determine f according to the initial value? i have looked at several books but they just assume this step is obvious

Last edited: Jun 26, 2012
2. Jun 26, 2012

### sunrah

have solved it now by taking a different approach and by converting to a system of ODEs and doing a coordinate transform from x(t) -> x(t,s) but I would still like to know how to solve it according to my original question, thanks