Thus x t lim xo f s x s ds Finally the validity of this integral equation has the following two implications X 1 x f s x s ds x 0 xg 1 x lim ff s x s ds x S lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function rx 1 satisfying the integral equation x 1 x f f s x s ds for all 1 t b 1 b 2 dx dt 0 ff s x s ds f 1 X 1 This now proves that the curve X 1 8 82 thus obtained is a solution of the initial value problem