Math 2.0

Ordinary differential equations Linear equations and systems Theorem

Liouville's formula

Builds on Fundamental matrixSystem of linear differential equations

Statement

Let II be an open interval, let A:I→ℝn×nA : I \to \mathbb{R}^{n \times n} be continuous, and let Φ:I→ℝn×n\Phi : I \to \mathbb{R}^{n \times n} be a differentiable function with Φ′(t)=A(t) Φ(t)\Phi'(t) = A(t)\, \Phi(t) for each t∈It \in I. Then for each t0,t∈It_0, t \in I, det⁡Φ(t)=det⁡Φ(t0) exp⁡(∫t0ttr⁡A(s) ds),\det \Phi(t) = \det \Phi(t_0)\, \exp\left( \int_{t_0}^{t} \operatorname{tr} A(s)\, ds \right), where tr⁡A(s)\operatorname{tr} A(s) is the trace of A(s)A(s).

Remarks

So det⁡Φ(t)\det \Phi(t) is zero at each point of II or at no point of II. This shows that a fundamental matrix is invertible everywhere if it is invertible at one point. For a constant matrix AA, the formula gives det⁡etA=ettr⁡A\det e^{tA} = e^{t \operatorname{tr} A}. For the system that comes from y′′+py′+qy=0y'' + p y' + q y = 0, the trace is −p-p, and the formula is Abel's identity.

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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…definitionOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionHomogeneous linear equation, definition, no evidence levelHomogeneous linear eq…definitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionSuperposition principle, statement, no evidence levelSuperposition princip…statementSolution space of a homogeneous linear equation has dimension n, statement, no evidence levelSolution space of a h…statementFundamental system of solutions, definition, no evidence levelFundamental system of…definitionFundamental matrix, definition, no evidence levelFundamental matrixdefinitionLiouville's formula, statement, no evidence levelLiouville's formulastatement
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Order of a differential equation definition
  6. Linear differential equation of order n definition
  7. Homogeneous linear equation definition
  8. System of linear differential equations definition
  9. Superposition principle statement
  10. Solution space of a homogeneous linear equation has dimension n statement
  11. Fundamental system of solutions definition
  12. Fundamental matrix definition
  13. Liouville's formula statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Liouville%27s_formula, Statement of Liouville's formula
About this page
Tag09PS3S0EF6
URIm2://09PS3S0EF6
Aliasesabel-jacobi-liouville-identity
Kindstatement, theorem
WritingW0 stub
MSC34A30
StatusThis copy of the site has no build data. Run m2 status 09PS3S0EF6.
UsesFundamental matrix, System of linear differential equations
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