I have a PDE problem set due on monday and as I look at the teacher's method for solving this problem it doesn't make any sense to me. I'll show what I have so far. Find the solution of [tex] U^2U_x + U_y = 0 [/tex] that satisfies [tex]u(x,0)=x[/tex] So I started out with the characteristic equations the way my professor did it in his notes [tex]x_t = u^2[/tex] [tex]y_t = 1 [/tex] [tex]z_t = 1[/tex] then integrating these I get [tex]t = \int 1/u^2 dx[/tex] [tex]t = y + y_0 [/tex] [tex]t = z + z_o[/tex] I don't know how to integrate the u in the first equation (as its a function of x - and I don't think that integrating w.r.t t will help.) and using the intial conditions [tex] x_o = s, y_o = 0, z_0 = s [/tex] so that reduces them to [tex]t = \int 1/u^2 dx[/tex] [tex]t = y [/tex] [tex]t = z + s[/tex] and this is where I got stuck. I know I have to solve for x and y as functions of s and t and substitute them into the third equation to get the solution but I can't figure out how to do this while eliminating the s and t (I mean I could solve for s if I could do the integral in one but I can't.) Anyways, all help would be greatly appreciated.