Systems of Linear ODEs

Systems of first-order linear ODEs arise naturally from higher-order equations, coupled oscillators, and compartmental models. The matrix-vector formulation unifies the theory.

Matrix Formulation

Definition5.1Linear system

A first-order linear system of $n$ equations is

$\mathbf{x}' = A(t)\mathbf{x} + \mathbf{g}(t)$

where $\mathbf{x}(t) \in \mathbb{R}^n$ , $A(t)$ is an $n \times n$ matrix, and $\mathbf{g}(t) \in \mathbb{R}^n$ . The system is homogeneous if $\mathbf{g} = \mathbf{0}$ and has constant coefficients if $A$ is constant.

RemarkReduction from higher order

Any $n$ -th order ODE $y^{(n)} + p_{n-1}y^{(n-1)} + \cdots + p_0 y = g(t)$ is equivalent to a first-order system: let $x_1 = y, x_2 = y', \ldots, x_n = y^{(n-1)}$ . Then $\mathbf{x}' = A\mathbf{x} + \mathbf{g}$ with companion matrix $A$ .

Fundamental Matrix

Definition5.2Fundamental matrix

A fundamental matrix for $\mathbf{x}' = A(t)\mathbf{x}$ is an $n \times n$ matrix $\Phi(t)$ whose columns are $n$ linearly independent solutions. The general solution is $\mathbf{x}(t) = \Phi(t)\mathbf{c}$ for $\mathbf{c} \in \mathbb{R}^n$ .

The matrix exponential fundamental matrix is $\Phi(t) = e^{At}$ (for constant $A$ ), defined by $e^{At} = \sum_{k=0}^\infty \frac{(At)^k}{k!}$ .

Theorem5.1Properties of the matrix exponential

For a constant matrix $A$ :

$e^{A \cdot 0} = I$ (identity matrix).
$(e^{At})' = Ae^{At} = e^{At}A$ .
$e^{(A+B)t} = e^{At}e^{Bt}$ if and only if $AB = BA$ .
$(e^{At})^{-1} = e^{-At}$ .
$\det(e^{At}) = e^{\text{tr}(A)t}$ .

Eigenvalue Method

ExampleSolving via eigenvalues

Solve $\mathbf{x}' = \begin{pmatrix}1 & 3\\3 & 1\end{pmatrix}\mathbf{x}$ .

Eigenvalues: $\det(A - \lambda I) = (1-\lambda)^2 - 9 = 0 \implies \lambda = 4, -2$ .

For $\lambda_1 = 4$ : $(A-4I)\mathbf{v} = 0 \implies \mathbf{v}_1 = \begin{pmatrix}1\\1\end{pmatrix}$ .

For $\lambda_2 = -2$ : $(A+2I)\mathbf{v} = 0 \implies \mathbf{v}_2 = \begin{pmatrix}1\\-1\end{pmatrix}$ .

General solution: $\mathbf{x}(t) = c_1 e^{4t}\begin{pmatrix}1\\1\end{pmatrix} + c_2 e^{-2t}\begin{pmatrix}1\\-1\end{pmatrix}$ .

ExampleComplex eigenvalues

Solve $\mathbf{x}' = \begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}\mathbf{x}$ .

Eigenvalues: $\lambda = \pm i$ . For $\lambda = i$ : $\mathbf{v} = \begin{pmatrix}1\\-i\end{pmatrix}$ .

Complex solution: $e^{it}\begin{pmatrix}1\\-i\end{pmatrix} = \begin{pmatrix}\cos t + i\sin t\\-i\cos t + \sin t\end{pmatrix} = \begin{pmatrix}\cos t\\\sin t\end{pmatrix} + i\begin{pmatrix}\sin t\\-\cos t\end{pmatrix}$ .

Real solution: $\mathbf{x}(t) = c_1\begin{pmatrix}\cos t\\\sin t\end{pmatrix} + c_2\begin{pmatrix}\sin t\\-\cos t\end{pmatrix}$ .

Nonhomogeneous Systems

Theorem5.2Variation of parameters for systems

The solution to $\mathbf{x}' = A\mathbf{x} + \mathbf{g}(t)$ , $\mathbf{x}(t_0) = \mathbf{x}_0$ is

$\mathbf{x}(t) = e^{A(t-t_0)}\mathbf{x}_0 + \int_{t_0}^t e^{A(t-s)}\mathbf{g}(s)\,ds.$

RemarkDuhamel's principle

The integral $\int_{t_0}^t e^{A(t-s)}\mathbf{g}(s)\,ds$ is a superposition of impulse responses, each weighted by $\mathbf{g}(s)$ and propagated from time $s$ to time $t$ by $e^{A(t-s)}$ . This is Duhamel's principle.