The following theorem tells us that solutions to first-order differential equations exist and are unique under certain reasonable conditions. Theorem 1.6.1. Existence and Uniqueness Theorem. Let x' = f(t, x) have the initial condition (to) = 0. If f and Of/Ox are continuous functions on the rectangle R={(t, x): 0t to|a, 0 x-xo|b}, there exists a unique solution u = u(t) for x' = f(t, x) and x(to) = x0 on some interval t-tol