Systems of Differential Equations

Many physical systems involve coupled variables. A system of two first-order ODEs:

$\frac{d\vec{x}}{dt} = A\vec{x}, \qquad \vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$

Solution method — Find eigenvalues $\lambda_i$ and eigenvectors $\vec{v}_i$ of $A$ :

$\vec{x}(t) = C_1\,e^{\lambda_1 t}\,\vec{v}_1 + C_2\,e^{\lambda_2 t}\,\vec{v}_2$

Phase portrait classification

Eigenvalues	Type	Stability
$\lambda_1, \lambda_2 < 0$ real	Node	Stable
$\lambda_1, \lambda_2 > 0$ real	Node	Unstable
$\lambda_1 < 0 < \lambda_2$ real	Saddle	Unstable
$\alpha \pm \beta i$ , $\alpha < 0$	Spiral	Stable
$\alpha \pm \beta i$ , $\alpha > 0$	Spiral	Unstable
$\pm \beta i$ (pure imaginary)	Centre	Neutrally stable

Example — Coupled spring-mass system with two masses yields a $4\times 4$ system. The eigenvalues give the normal mode frequencies.

import numpy as np
A = np.array([[-3, 1], [2, -2]])
vals, vecs = np.linalg.eig(A)
print("Eigenvalues:", vals)
print("Eigenvectors:\n", vecs)

Key insight: The eigenvalues of the system matrix completely determine the long-term behaviour — stability, oscillation, and decay rate.

Common pitfall: In coupled systems, each equation’s rate depends on the other variable — solving them one at a time as if independent destroys exactly the interaction the system exists to model.

Placeholder: Systems of Differential Equations

Practise this lesson

Differential Equations and Vector Calculus