Euler Darboux PDE solution

Question

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$

What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I can use that to find a solution for $k>0$. Hints and any help is appreciated.

Thanks, felasfa

Answer 1

Let $$ \xi = x-y, \\ \eta = x+y, $$ then $$ u_x - u_y = 2u_{\xi}, \quad u_{xy} = u_{\eta\eta}- u_{\xi\xi}. $$ The equation becomes: $$ u_{\eta\eta}- u_{\xi\xi} + \frac{2k}{\xi} u_{\xi}= 0.\tag{1} $$ Further let $t = \xi^{\alpha}$ ($\alpha\neq -1$), then $$ u_{\xi} = \alpha \xi^{\alpha-1}u_t ,\quad u_{\xi\xi} =\alpha^2 \xi^{2\alpha-2} u_{tt}+ \alpha(\alpha-1)\xi^{\alpha-2}u_t, $$ plugging back to (1) gives: $$ u_{\eta\eta} - \alpha^2 \xi^{2\alpha-2} u_{tt}- \alpha(\alpha-1)\xi^{\alpha-2}u_t +2k\alpha \xi^{\alpha-2} u_t = 0. $$ Let $\alpha = 2k+1>1$, first order terms get canceled: $$ u_{\eta\eta} - \alpha^2 t^{\frac{2(\alpha-1)}{\alpha} } u_{tt} = 0. $$ Rewrite this as: $$ u_{tt} - \frac{1}{(2k+1)^2}t^{\frac{-4k}{2k+1}} u_{\eta\eta} = 0.\tag{2} $$ (2) is the Tricomi-type wave equation describing some quantity's transition from subsonic flow (elliptic region) to supersonic flow (hyperbolic region), you can try your luck with separation of variable $$u = F(t)G(\eta) = F\Big((x-y)^{2k+1}\Big)G(x+y).$$