Variation of constants formula for linear systems

Builds on System of linear differential equations Fundamental matrix Matrix exponential as the solution of a linear system Variation of parameters

Statement

Let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ and $b : I \to \mathbb{R}^n$ be continuous, and let $\Phi$ be a fundamental matrix of $x' = A(t) x$ on $I$ . Let $t_0 \in I$ and $x_0 \in \mathbb{R}^n$ . Then the unique solution on $I$ of $x' = A(t) x + b(t)$ , $x(t_0) = x_0$ is $x(t) = \Phi(t)\, \Phi(t_0)^{-1} x_0 + \Phi(t) \int_{t_0}^{t} \Phi(s)^{-1}\, b(s)\, ds, \qquad t \in I.$

Remarks

For a constant matrix $A$ , take $\Phi(t) = e^{tA}$ . The formula becomes $x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^{t} e^{(t - s) A}\, b(s)\, ds$ . The formula comes from the trial solution $x(t) = \Phi(t)\, c(t)$ : the equation gives $\Phi(t)\, c'(t) = b(t)$ . This is variation of parameters for systems. For $n = 1$ , it is the solution formula for linear first-order equations.

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Differential equation concept
Function concept
Interval concept
Wronskian definition
Ordinary differential equation definition
Order of a differential equation definition
Solution of an ordinary differential equation definition
Initial value problem definition
Linear differential equation of order n definition
System of linear differential equations definition
Homogeneous linear equation definition
Homogeneous linear system with constant coefficients definition
Superposition principle statement
Matrix exponential as the solution of a linear system statement
Solution space of a homogeneous linear equation has dimension n statement
Fundamental system of solutions definition
Fundamental matrix definition
General solution of an inhomogeneous linear equation statement
Variation of parameters concept
Variation of constants formula for linear systems statement

Tag	`0S1KG0XE40`
URI	`m2://0S1KG0XE40`
Aliases	`duhamel-formula`
Kind	statement, theorem
Writing	W0 stub
MSC	34A30
Status	This copy of the site has no build data. Run `m2 status 0S1KG0XE40`.
Uses	System of linear differential equations, Fundamental matrix, Matrix exponential as the solution of a linear system, Variation of parameters

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