I was going through an inroductory book on PDE's and at one point they proceed with little show of work. I have problem with equation [itex] -yu_x + xu_y = u [/itex]. Characteristics for this equation are [itex] x_t = -y, y_t = x, u_t = u [/itex]. So far it is clear, but now books states that solution of first characteristic is [itex] x(t,s) = f_1(s)sin(t) + f_2(s)cos(t) [/itex], which is perplexing to me, I would just integrate righthand side treating x or y as constants (we are integrating with respect to t). Any suggestion?