Let $F:[-a, a] \times\left[x_{0}-r, x_{0}+r\right] \rightarrow \mathbf{R}$ be a continuous function. Let $C$ be the maximum value of $|F(t, x)|$. Suppose there is a constant $K$ such that

$$|F(t, x)-F(t, y)| \leqslant K|x-y|$$

for all $t \in[-a, a]$ and $x, y \in\left[x_{0}-r, x_{0}+r\right]$. Let $b<\min (a, r / C, 1 / K)$. Show that there is a unique $C^{1}$ function $x:[-b, b] \rightarrow\left[x_{0}-r, x_{0}+r\right]$ such that

$$x(0)=x_{0}$$

and

$$\frac{d x}{d t}=F(t, x(t)) .$$

[Hint: First show that the differential equation with its initial condition is equivalent to the integral equation

$$\left.x(t)=x_{0}+\int_{0}^{t} F(s, x(s)) d s .\right]$$