System of first-order linear ODEs

A system of first-order linear ODEs describes multiple coupled functions whose rates of change depend on each other. Written in matrix form:

${\mathbf{x}}^{\prime }=A(t)\mathbf{x}+\mathbf{b}(t)$

where $\mathbf{x}(t)\in {\mathbb{R}}^n$ is the unknown vector function, $A(t)$ is an $n\times n$ matrix of coefficients, and $\mathbf{b}(t)$ is a vector of forcing terms.

If $\mathbf{b}(t)=0$, the system is homogeneous: ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$.

If $A$ is constant (doesn’t depend on $t$), the system has constant coefficients, the most common case in introductory ODE courses.

How systems arise

Two common ways:

1. Multiple interacting quantities, each with its own rate equation, all coupled. Examples:

   • Population dynamics: prey and predator populations affect each other (Lotka-Volterra).
   • Mixing problems: connected tanks where solute flows between them.
   • Mass-spring systems with multiple masses connected by springs.

2. Reducing a higher-order ODE — any $n$-th order ODE can be rewritten as $n$ first-order ODEs by introducing intermediate variables.

For an $n$-th order ODE $y^{(n)}=F(t,y,y^{\prime },\dots ,y^{(n-1)})$, set:

$x_1=y,x_2=y^{\prime },\dots ,x_n=y^{(n-1)}$

Then the original ODE becomes the system:

$\{\begin{array}{l}{x}_1^{\prime }=x_2\\ x_2^{\prime }=x_3\\ ⋮\\ x_n^{\prime }=F(t,x_1,x_2,\dots ,x_n)\end{array}$

With $\mathbf{x}=(x_1,\dots ,x_n)$ and a suitable $A$ and $\mathbf{b}$, this is in standard form.

So we only need solution methods for first-order systems. Higher-order ODEs are a special case.

Two-dimensional example

A common $2\times 2$ constant-coefficient system:

$\{\begin{array}{l}{x}_1^{\prime }=a{x}_1+b{x}_2\\ x_2^{\prime }=c{x}_1+d{x}_2\end{array}$

In matrix form:

$\frac{d}{dt}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\Rightarrow {\mathbf{x}}^{\prime }=A\mathbf{x}$

Solution via eigenvalues

For a constant-coefficient homogeneous system ${\mathbf{x}}^{\prime }=A\mathbf{x}$, try the ansatz $\mathbf{x}(t)=\mathbf{v}{e}^{\lambda t}$ where $\mathbf{v}$ is a constant vector and $\lambda$ a scalar.

Substituting: $\lambda \mathbf{v}{e}^{\lambda t}=A\mathbf{v}{e}^{\lambda t}$, which simplifies to

$A\mathbf{v}=\lambda \mathbf{v}$

So $\lambda$ is an eigenvalue of $A$ and $\mathbf{v}$ is the corresponding eigenvector. The eigenvalues are the roots of the characteristic polynomial:

$\det (A-\lambda I)=0$

For a $2\times 2$ matrix, this gives ${\lambda }^2-(a+d)\lambda +(ad-bc)=0$ — a quadratic, so up to two eigenvalues.

Each eigenvalue/eigenvector pair $({\lambda }_i,{\mathbf{v}}_i)$ contributes one solution ${\mathbf{x}}_i(t)={\mathbf{v}}_i{e}^{{\lambda }_{i}t}$. The general solution is a linear combination:

$\mathbf{x}(t)=c_1{\mathbf{v}}_1{e}^{{\lambda }_{1}t}+c_2{\mathbf{v}}_2{e}^{{\lambda }_{2}t}+\cdots$

This is the matrix analog of the Characteristic equation for second-order ODEs.

Three cases by eigenvalue type

Just as for scalar second-order ODEs, the behavior depends on what the eigenvalues look like:

• Distinct real eigenvalues: solution is a linear combination of exponentials with distinct rates. See Distinct real eigenvalues case.
• Complex conjugate eigenvalues $\alpha \pm i\beta$: solution involves $e^{\alpha t}\cos (\beta t)$ and $e^{\alpha t}\sin (\beta t)$. See Complex conjugate eigenvalues case.
• Repeated eigenvalues: need generalized eigenvectors. See Repeated eigenvalues case.

Related: Existence and uniqueness for systems, Linear independence of vector functions for when the homogeneous solution lacks enough free parameters, and Phase plane behaviour for trajectories of 2D systems.