First order differential equation by the method of characteristics

Question

Solve $-yU_x + xU_y= U$, initial conditions $U(x,0)= P(x)$. Characteristic equations: $X_t =-Y$, $Y_t=X$, $U_t=U$. The book says $X(t,s) =F_1(s)\cos t + F_2(s)\sin t$, $Y(t,s)=F_1(s)\sin t -F_2(s)\cos t$. I can do $U(t,s)$. I can't work out how $X$ and $Y$ were done.