Maximal Interval of existence for the Given Initial value problem.

Question

The given initial value problem is

$$ y{\prime}(t)=f(y(t))\;\;,\;\;y(0)=a \in\mathbb{R}\\where\;\; f:\mathbb{R}\to \mathbb{R}$$

Then

$\text{The maximal interval of existence for the above problem is $\mathbb{R}$ when $f$ is bounded and} $ $\text{continuously differentiable}$

i have no idea how to proceed . Please help

Thankyou.

Answer 1

If $f$ is bounded, $|f(y)| \le M$ for all $y\in\Bbb R$, then you get for the solution if the IVP $$ |y(t)-a|\le M\,|t| $$ for all $t$ in the domain of the maximal solution.

As the maximal solution $y$ can not reach infinity at finite times, it is defined on the whole of $\Bbb R$.