Riemann function for a second order hyperbolic PDE

[Kate2010]

Homework Statement

Find the Riemann function for

u_xy + xyu_x = 0, in x + y > 0
u = x, u_y = 0, on x+y = 0

Homework Equations

The Attempt at a Solution

I think the Riemann function, R(x,y;s,n), must satisfy:

0 = R_xy - (xyR)_x
R_x = 0 on y =n
R_y = xyR on x = s
R = 1 at (x,y) = (s,n)

But I don't know how to solve this beyond just spotting a solution.

 

[I like Serena]

Homework Statement

Find the Riemann function for

u_xy + xyu_x = 0, in x + y > 0
u = x, u_y = 0, on x+y = 0

Homework Equations

The Attempt at a Solution

I think the Riemann function, R(x,y;s,n), must satisfy:

0 = R_xy - (xyR)_x
R_x = 0 on y =n
R_y = xyR on x = s
R = 1 at (x,y) = (s,n)

But I don't know how to solve this beyond just spotting a solution.

How about:
0 = R_xy - (xyR)_x
g(y) = R_y - xyR

Pick g(y)=0, just to find a particular solution.
R_y = xyR
R_y/R = xy
ln R = xy²/2 + h(x)
R=exp(xy²/2) H(x)

 

[Kate2010]

Thanks :)