Math 2.0

Ordinary differential equations Example

Blow-up in finite time

Builds on Maximal solutionSeparable equationInitial value problem

Example

Let x0>0x_0 > 0 and consider the initial value problem x′(t)=x(t)2x'(t) = x(t)^2, x(0)=x0x(0) = x_0. Separation of variables gives the solution x(t)=x01−x0t,t∈(−∞,1x0).x(t) = \frac{x_0}{1 - x_0 t}, \qquad t \in \left(-\infty, \frac{1}{x_0}\right). This is the maximal solution: x(t)→∞x(t) \to \infty as t→1/x0t \to 1/x_0 from below. So the solution blows up at the finite time 1/x01/x_0, although the field f(x)=x2f(x) = x^2 is smooth on all of ℝ\mathbb{R}.

Remarks

The blow-up time 1/x01/x_0 depends on the initial value. Larger initial values blow up sooner. For x0<0x_0 < 0, the maximal interval is (1/x0,∞)(1/x_0, \infty), and for x0=0x_0 = 0 the solution is 00 on ℝ\mathbb{R}. The field x2x^2 is not globally Lipschitz. A globally Lipschitz field has no blow-up. Near the blow-up time, x(t)x(t) behaves like (1/x0−t)−1(1/x_0 - t)^{-1}. This is a finite-time singularity.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSeparable equation, definition, no evidence levelSeparable equationdefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinitionBlow-up in finite time, example, no evidence levelBlow-up in finite timeexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Separable equation definition
  8. Solution of an ordinary differential equation definition
  9. Initial value problem definition
  10. Uniqueness of solutions for Lipschitz vector fields statement
  11. Maximal solution definition
  12. Blow-up in finite time example

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinary_differential_equation, Global uniqueness and maximum domain of solution
  2. Web https://en.wikipedia.org/wiki/Singularity_(mathematics), Finite-time singularity
About this page
Tag0E4GXX7SJP
URIm2://0E4GXX7SJP
Aliasesfinite-time-blow-up
Kindexample
WritingW0 stub
MSC34C11
Instance ofMaximal solution
UsesMaximal solution, Separable equation, Initial value problem

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