---
tag: 0E4GXX7SJP
kind: example
name: Blow-up in finite time
aliases: [finite-time-blow-up]
msc: [34C11]
instance_of: [01J7FZ65J0]
uses: [019VSXSFMQ, 0V4E623THY, 075VBY4K4X]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinary_differential_equation
    locator: Global uniqueness and maximum domain of solution
  - url: https://en.wikipedia.org/wiki/Singularity_(mathematics)
    locator: Finite-time singularity
---

## Example

Let $x_0 > 0$ and consider the [[initial value problem|075VBY4K4X]] $x'(t) = x(t)^2$, $x(0) = x_0$.
[[Separation of variables|0V4E623THY]] gives the solution
$$x(t) = \frac{x_0}{1 - x_0 t}, \qquad t \in \left(-\infty, \frac{1}{x_0}\right).$$
This is the [[maximal solution|019VSXSFMQ]]: $x(t) \to \infty$ as $t \to 1/x_0$ from below.
So the solution blows up at the finite time $1/x_0$, although the field $f(x) = x^2$ is smooth on all of $\mathbb{R}$.

## Remarks

The blow-up time $1/x_0$ depends on the initial value. Larger initial values blow up sooner.
For $x_0 < 0$, the maximal interval is $(1/x_0, \infty)$, and for $x_0 = 0$ the solution is $0$ on $\mathbb{R}$.
The field $x^2$ is not globally Lipschitz. A [[globally Lipschitz field|0BFD3A6CRH]] has no blow-up.
Near the blow-up time, $x(t)$ behaves like $(1/x_0 - t)^{-1}$. This is a finite-time singularity.

## Explanations

### L1 human

The rate of growth is the square of the size, so the growth feeds itself very fast.
The solution reaches infinity in finite time, unlike exponential growth.