Math 2.0

Ordinary differential equations Linear equations and systems Theorem

Duhamel's principle

Builds on System of linear differential equationsFundamental matrixInitial value problemMatrix exponentialVariation of parameters

Statement

Let II be an open interval, and let A ⁣:I→ℝn×nA \colon I \to \R^{n \times n} and f ⁣:I→ℝnf \colon I \to \R^n be continuous. Let Φ\Phi be a fundamental matrix of the homogeneous linear system x′=A(t)xx' = A(t) x on II, and put Φ(t,s)=Φ(t)Φ(s)−1\Phi(t, s) = \Phi(t) \Phi(s)^{-1} for t,s∈It, s \in I. Let t0∈It_0 \in I and x0∈ℝnx_0 \in \R^n. Then the initial value problem x′=A(t)x+f(t),x(t0)=x0x' = A(t) x + f(t), \qquad x(t_0) = x_0 has exactly one solution on II, and it is x(t)=Φ(t,t0) x0+∫t0tΦ(t,s) f(s) ds.x(t) = \Phi(t, t_0)\, x_0 + \int_{t_0}^{t} \Phi(t, s)\, f(s)\, ds. If AA is constant, then Φ(t,s)=e(t−s)A\Phi(t, s) = e^{(t - s) A}, the matrix exponential.

Remarks

Differentiating under the integral sign gives x′(t)=A(t)Φ(t,t0)x0+Φ(t,t)f(t)+∫t0tA(t)Φ(t,s)f(s) ds=A(t)x(t)+f(t)x'(t) = A(t) \Phi(t, t_0) x_0 + \Phi(t, t) f(t) + \int_{t_0}^t A(t) \Phi(t, s) f(s)\, ds = A(t) x(t) + f(t), since Φ(t,t)\Phi(t, t) is the identity. Uniqueness holds because the difference of two solutions solves the homogeneous system with initial value 00, so it is zero. The formula is the matrix form of variation of parameters.

Example: for x′=−x+1x' = -x + 1 and x(0)=0x(0) = 0, one has Φ(t,s)=e−(t−s)\Phi(t, s) = e^{-(t - s)}. The formula gives x(t)=∫0te−(t−s) ds=1−e−tx(t) = \int_0^t e^{-(t - s)}\, ds = 1 - e^{-t}, and indeed x′=e−t=−x+1x' = e^{-t} = -x + 1.

Going deeper

For a student

The function Φ(t,s)\Phi(t, s) is often called the state transition matrix or propagator. The same idea works for evolution equations such as the heat equation and the wave equation, where Φ(t,s)\Phi(t, s) becomes a solution operator. In the theory of C0C_0-semigroups, the formula defines the mild solution of an inhomogeneous abstract Cauchy problem. For a boundary value problem, the analogous object is the Green's function.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptFundamental matrix, concept, no evidence levelFundamental matrixconceptInitial value problem, concept, no evidence levelInitial value problemconceptInterval, concept, no evidence levelIntervalconceptMatrix exponential, concept, no evidence levelMatrix exponentialconceptSystem of linear differential equations, concept, no evidence levelSystem of linear diff…conceptWronskian, definition, no evidence levelWronskiandefinitionOrdinary 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…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…definitionGeneral solution of an inhomogeneous linear equation, statement, no evidence levelGeneral solution of a…statementVariation of parameters, concept, no evidence levelVariation of paramete…conceptDuhamel's principle, statement, no evidence levelDuhamel's principlestatement
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. Fundamental matrix concept
  4. Initial value problem concept
  5. Interval concept
  6. Matrix exponential concept
  7. System of linear differential equations concept
  8. Wronskian definition
  9. Ordinary differential equation definition
  10. Order of a differential equation definition
  11. Linear differential equation of order n definition
  12. Homogeneous linear equation definition
  13. Superposition principle statement
  14. Solution space of a homogeneous linear equation has dimension n statement
  15. Fundamental system of solutions definition
  16. General solution of an inhomogeneous linear equation statement
  17. Variation of parameters concept
  18. Duhamel's principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Duhamel%27s_principle
About this page
TagQ28DM6ZCMF
URIm2://Q28DM6ZCMF
Kindstatement, theorem
WritingW0 stub
MSC34A30, 34A12
StatusThis copy of the site has no build data. Run m2 status Q28DM6ZCMF.
AboutSystem of linear differential equations
UsesSystem of linear differential equations, Fundamental matrix, Initial value problem, Matrix exponential, Variation of parameters
Changed2 November 2026

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