Variation of Parameters for Systems

Variation of parameters provides a general method for solving nonhomogeneous linear systems once the fundamental matrix of the homogeneous system is known.

Statement

Theorem5.8Variation of parameters for systems

Consider $\mathbf{x}' = A(t)\mathbf{x} + \mathbf{g}(t)$ with fundamental matrix $\Phi(t)$ for the homogeneous system. A particular solution is

$\mathbf{x}_p(t) = \Phi(t)\int_{t_0}^t \Phi^{-1}(s)\mathbf{g}(s)\,ds.$

The general solution is $\mathbf{x}(t) = \Phi(t)\mathbf{c} + \Phi(t)\int_{t_0}^t \Phi^{-1}(s)\mathbf{g}(s)\,ds$ .

Derivation

Proof

Assume $\mathbf{x}_p = \Phi(t)\mathbf{u}(t)$ for an unknown vector $\mathbf{u}(t)$ . Substituting into the nonhomogeneous equation:

$\mathbf{x}_p' = \Phi'(t)\mathbf{u}(t) + \Phi(t)\mathbf{u}'(t) = A\Phi(t)\mathbf{u}(t) + \Phi(t)\mathbf{u}'(t).$

Since $\mathbf{x}_p' = A\mathbf{x}_p + \mathbf{g} = A\Phi\mathbf{u} + \mathbf{g}$ :

$A\Phi\mathbf{u} + \Phi\mathbf{u}' = A\Phi\mathbf{u} + \mathbf{g} \implies \Phi(t)\mathbf{u}'(t) = \mathbf{g}(t).$

Therefore $\mathbf{u}'(t) = \Phi^{-1}(t)\mathbf{g}(t)$ , and integrating gives $\mathbf{u}(t) = \int_{t_0}^t\Phi^{-1}(s)\mathbf{g}(s)\,ds$ . $\blacksquare$

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Examples

ExampleVariation of parameters example

Solve $\mathbf{x}' = \begin{pmatrix}2&-1\\3&-2\end{pmatrix}\mathbf{x} + \begin{pmatrix}0\\t\end{pmatrix}$ .

Eigenvalues: $\lambda = 1, -1$ . Fundamental matrix: $\Phi = \begin{pmatrix}e^t & e^{-t}\\e^t & 3e^{-t}\end{pmatrix}$ .

$\Phi^{-1} = \frac{1}{2}\begin{pmatrix}3e^{-t} & -e^{-t}\\-e^t & e^t\end{pmatrix}$ .

$\Phi^{-1}\mathbf{g} = \frac{1}{2}\begin{pmatrix}-te^{-t}\\te^t\end{pmatrix}$ .

Integrating and multiplying by $\Phi$ gives the particular solution.

RemarkConstant coefficients simplification

For constant $A$ , the formula simplifies to $\mathbf{x}_p(t) = \int_{t_0}^t e^{A(t-s)}\mathbf{g}(s)\,ds$ , which is the convolution of the matrix exponential with the forcing term. This connects to the transfer function approach via Laplace transforms.

ExampleImpulse response matrix

The impulse response matrix (or Green's function) for $\mathbf{x}' = A\mathbf{x}$ is $G(t,s) = \Phi(t)\Phi^{-1}(s)$ for $t \geq s$ . For constant $A$ : $G(t,s) = e^{A(t-s)}$ . The response to $\mathbf{g}(t) = \mathbf{b}\delta(t-\tau)$ is $\mathbf{x}(t) = e^{A(t-\tau)}\mathbf{b}$ for $t > \tau$ .