Suppose we have a first order linear PDE of the form: a(x,y) ux + b(x,y) uy = 0 Then dy/dx = b(x,y) / a(x,y) [assumption: a(x,y) is not zero] The characteristic equation for the PDE is b(x,y) dx - a(x,y) dy=0 d[F(x,y)]=0 "F(x,y)=constant" are characteristic curves Therefore, the general solution to the PDE is u(x,y)=f[F(x,y)] where f is an arbitrary function. =========================================== I don't understand the parts in red. 1) Why would dy/dx = b(x,y) / a(x,y) ? This doesn't seem obvious to me at all...how can we derive (or prove) it? 2) Also, what is the meaning of the equation d[F(x,y)]=0? Thanks for explaining!