Find the Riemann function for the PDE $u_{xy} + xyu_x = f(x,y)$ and use it to show the solution to the problem
$u_{xy} + xyu_x = 0$ in $x+y>0$
$u = x, u_y = 0$ on $x+y=0$
is $u(\xi, \eta) = -\eta + \int_{-\eta}^{\xi} exp(x(x^2 - \eta^2)/2) \,dx$
So far I have worked out that if the Riemann function R satisfies:
$R_{xy} -(xyR)_x = 0$ for all $x <\xi, y<\eta$
$R_x = 0$ on $y=\eta$
$R_y = xyR$ on $x = \xi$
$R=1$ at $(x,y)=(\xi, \eta) $
then the PDE has solution
$u(\xi, \eta) = \iint\limits_{D}Rf dxdy + R(B)u(B) - \int_{A}^{B}R(u_y +uxy)dy + uR_x dx$
where D is the triangle with vertices $(\xi, \eta), B(\xi, -\xi), A(-\eta, \eta)$
I also suspect $R$ involves $H(\xi-x)H(\eta-y)$, but am struggling to proceed to find $R$ that satisfies all 4 conditions above. Any tips would be much appreciated, thank you!