Systems of Differential Equations

Coupled Systems & Solutions

Many physical systems involve coupled variables where each rate depends on the other. Solving them one at a time destroys the interaction the system models.

A system of two first-order ODEs is written as $\frac{d\vec{x}}{dt} = A\vec{x}$, where $\vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$.

Solution method: Find eigenvalues $\lambda_i$ and eigenvectors $\vec{v}_i$ of matrix $A$ to form the general solution:

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

Key insight: The eigenvalues of $A$ completely determine long-term behaviour, stability, oscillation, and decay rates.

Phase Portraits & Example

The eigenvalues classify the phase portrait behaviour:

Worked Example: A coupled spring-mass system yields a $4 \times 4$ system. Its eigenvalues give the normal mode frequencies.

A = np.array([[-3, 1], [2, -2]])
vals, vecs = np.linalg.eig(A)