As I continue working through lecture notes for my DE course, I encounter the following as an exercise:
Looking at the PDE
$$e^{2y}u_{xx}+u_y=u_{yy}$$
how can we find the differential equation satisfied by its characteristic curves and show that $$\lambda =x+e^y \text{ and } \mu =x-e^y $$are canonical variables for the PDE?
Any help would be very appreciate. Best regards, MM