Here is an alternative proof that solutions are unique and have Lipschitz de- pendence upon initial conditions when f is Lipschitz. Suppose that u, v : ] → B(x) are two solutions of the ODE x = f(t,x), where f: J× B(x) → R has a uniformly Lipschitz dependence on x with constant K. We make no assumptions about the dependence of f q(t)=|u(t)—v(t)|². i=1 on t. Define (a) Use the inner product (u,v) = Σu;v; and the Schwarz inequality, |(u,v)|≤|u||v|for vectors in R", to find an ordinary differential inequal- ity for o, i.e., an equation of the form (t)≤F(t,q). -2Kt (b) Using this inequality show (ekt (t)) ≤0. Therefore, if t> to, show that dt |u(t)—v(t)|≤ e(t−t.)|u(t。)— v(t。)|. Conclude that the solution is unique and that two nearby solutions deviate at most exponentially in time.