Preview
IB DP Mathematics: Applications and Interpretation | HL | Unit 5: Calculus | Topic 5.16
📋 Learning Intentions
- Write a system of two coupled first-order linear differential equations in matrix form \( \mathbf{x}' = A\mathbf{x} \).
- Find eigenvalues and eigenvectors of a 2×2 matrix and interpret their role in the general solution.
- State the general solution of a coupled system using eigenvalues and eigenvectors, including real distinct, repeated, and complex cases.
- Solve initial value problems for coupled differential equations by applying initial conditions to determine arbitrary constants.
- Sketch phase portraits and interpret the long-term behaviour of solutions from the signs of eigenvalues.
- Apply coupled differential equation models to real-world contexts such as population dynamics and compartmental models.
Real-World Introduction
Consider a predator–prey ecosystem: the population of rabbits grows when foxes are scarce, but declines when foxes are abundant. The fox population grows when rabbits are plentiful, but declines when food is scarce. Neither population can be modelled in isolation — each rate of change depends on the other variable simultaneously.
This mutual dependence is captured by a coupled system of differential equations. The same mathematical structure appears in pharmacokinetics (drug moving between blood and tissue compartments), electrical circuits (current and charge interacting), and epidemiology (susceptible and infected populations). Eigenvalue analysis transforms what appears to be an intractable pair of equations into a clean, interpretable solution.
🌍 Why This Matters Epidemiologists used coupled differential equation models (SIR/SIS compartmental models) during the COVID-19 pandemic to forecast infection peaks and evaluate intervention timing. The eigenvalues of the system matrix directly determine whether an outbreak grows or decays — a result with immediate policy consequences.
Core Concepts & Definitions
A single first-order ODE such as \( \frac{dx}{dt} = kx \) has one unknown function. When two quantities each influence the other's rate of change, we need two equations simultaneously. These are called coupled because the derivative of each variable depends on both variables.
📖 Key Definition — Coupled System of Differential Equations A coupled system of first-order linear differential equations has the form: \[ \frac{dx}{dt} = ax + by, \qquad \frac{dy}{dt} = cx + dy \] where \(a, b, c, d\) are real constants and \(x(t), y(t)\) are unknown functions of \(t\). The system is coupled because each equation contains both \(x\) and \(y\).
The power of matrix algebra is that this system can be written compactly. Defining the state vector \(\mathbf{x}(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}\) and the coefficient matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the system becomes \(\mathbf{x}' = A\mathbf{x}\). This single matrix equation is the starting point for all solution methods.
Matrix Form of a Coupled System
\[ \frac{d\mathbf{x}}{dt} = A\mathbf{x}, \qquad \text{where } \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}, \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
Eigenvalues and Eigenvectors of the Coefficient Matrix
The solution strategy is to look for solutions of the form \(\mathbf{x}(t) = e^{\lambda t}\mathbf{v}\), where \(\lambda\) is a scalar and \(\mathbf{v}\) is a constant vector.