Exercise 13

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

1. Define $$\phi(t) = \int_{t_0}^t |u(s) - v(s)| \, ds,$$ then $\phi(t_0) = 0$ and $\phi(t) \geq 0$ for $t \geq t_0$. Show that $$\phi'(t) = |u(t) - v(t)|.$$

2. Since $$|u(t) - v(t)| - K \int_{t_0}^t |u(s) - v(s)| \, ds \leq 0,$$ we know that $$\phi'(t) - K \phi(t) \leq 0.$$ Use this fact to show that $$\frac{d}{dt} \left[ e^{-Kt} \phi(t) \right] \leq 0.$$ Conclude that $$\phi(t) = \int_{t_0}^t |u(s) - v(s)| \, ds = 0$$ for $t \geq t_0$ or for all $t \geq t_0$ and $u(t) = v(t)$.