Exercise 5

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

Use the Fundamental Theorem of Calculus to show that the function $u = u(t)$ is a solution to the initial value problem $$\begin{aligned} x' & = f(t, x) \\ x(t_0) & = x_0, \end{aligned}$$ if and only if $u$ is a solution to the integral equation $$x(t) = x_0 + \int_{t_0}^t f(s, x(s)) \, ds.$$