"Consider a first order linear PDE. (e.g. y ux + x uy = 0) If u(x,y) is constant along the curves y2 - x2 = c, then this implies that the general solution to the PDE is u(x,y) = f(y2 - x2) where f is an arbitrary differentiable funciton of one variable. We call the curves along which u(x,y) is constant the characteristic curves." =============================== I don't understand the implication above highlighted in red. The idea of characteristic curves seems to be very important in solving first order linear PDEs, but I am never able to completely understand the idea of it. Why would finding the characteristic curves help us find the general solution to the PDE? Thanks for explaining!