Exercise 12

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

To show uniqueness, assume that $u(t)$ and $v(t)$ are both solutions to $$x(t) = x_0 + \int_{t_0}^t f(s, x(s)) \, ds.$$ Show that $$|u(t) - v(t)|\leq K \int_{t_0}^t |u(s) - v(s)| \, ds.$$