8.2.2 Coupled First Order Linear Equations

Introduction to Coupled Differential Equations

Coupled first-order linear equations involve two or more dependent variables, such as $f(t)$ and $r(t)$ , that interact and influence each other.

These equations often model real-world situations such as predator-prey dynamics, where the populations of two species depend on one another over time.

The general form for a pair of coupled equations is:

\frac{df}{dt} = af + br, \quad \frac{dr}{dt} = cf + dr

where $f(t)$ and $r(t)$ are the dependent variables, and $a, b, c, d$ are constants.

To solve these equations:

Rewrite one equation as a second-order equation for one variable.
Solve the second-order equation.
Back-substitute to find the other variable.

Worked Example: Predator-Prey Dynamics

infoNote

Problem

In a population of foxes ( $f$ ) and rabbits ( $r$ ), the dynamics are governed by:

\frac{df}{dt} = 3r - 6f, \quad \frac{dr}{dt} = 4r - 8f

The initial populations are $f(0) = 2$ and $\frac{df}{dt}\big|_{t=0} = -2$

Find $f(t)$ , $r(t)$ , and the long-term populations of foxes and rabbits.

Step 1: Rewrite as a Second-Order Equation

Start with $\frac{df}{dt} = 3r - 6f$

Differentiate both sides with respect to $t$ :

\frac{d^2f}{dt^2} = 3\frac{dr}{dt} - 6\frac{df}{dt}

Substitute $\frac{dr}{dt} = 4r - 8f$

\frac{d^2f}{dt^2} = 3(4r - 8f) - 6\frac{df}{dt}

Substitute $r = \frac{df}{dt} + 6f$ from the first equation:

\frac{d^2f}{dt^2} = 3(4(\frac{df}{dt} + 6f) - 8f) - 6\frac{df}{dt}

Simplify:

\frac{d^2f}{dt^2} = 12\frac{df}{dt} + 72f - 24f - 6\frac{df}{dt}

\frac{d^2f}{dt^2} = 6\frac{df}{dt} + 48f

Step 2: Solve the Second-Order Equation

The second-order equation is:

\frac{d^2f}{dt^2} - 6\frac{df}{dt} - 48f = 0

Solve the characteristic equation:

m^2 - 6m - 48 = 0

Using the quadratic formula:

m = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-48)}}{2(1)}

= \frac{6 \pm \sqrt{36 + 192}}{2} = \frac{6 \pm 15}{2}

m = 10.5, \, m = -4.5

General solution for $f(t)$

f(t) = A e^{10.5t} + B e^{-4.5t}

Step 3: Use Initial Conditions to Solve for Constants

At $t = 0$ , $f(0) = 2$

2 = A e^{0} + B e^{0} \quad \Rightarrow \quad A + B = 2

At $t=0, \frac{df}{dt} = -2$

Differentiate $f(t)$

\frac{df}{dt} = 10.5A e^{10.5t} - 4.5B e^{-4.5t}

Substituting $t=0$

-2 = 10.5A - 4.5B

Solve the simultaneous equations:

A + B = 2, \quad 10.5A - 4.5B = -2.

Solve for $A$ and $B$ :

A = 0.5, \, B = 1.5

Final solution for $f(t)$

f(t) = 0.5e^{10.5t} + 1.5e^{-4.5t}

Step 4: Find $r(t)$

Use $r = \frac{df}{dt} + 6f$

\frac{df}{dt} = 10.5(0.5)e^{10.5t} - 4.5(1.5)e^{-4.5t}

r(t) = \left[5.25e^{10.5t} - 6.75e^{-4.5t}\right] + 6\left[0.5e^{10.5t} + 1.5e^{-4.5t}\right]

Simplify:

r(t) = 8.25e^{10.5t} + 1.25e^{-4.5t}

Step 5: Long-Term Behavior

As $t \to \infty$

$e^{10.5t} \to \infty$ , suggesting unbounded growth of $f(t)$ and $r(t)$ if no damping or external factors are applied.

Note Summary

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Common Mistakes

Confusing roles of variables: Be clear about dependent and independent variables.
Errors in substitution: Check carefully when replacing one variable in terms of another.
Mismanaging constants in solutions: Use initial conditions consistently to find $A$ and $B$ .
Skipping interpretation: Always analyze the physical or long-term meaning of the solution.

infoNote

Key Formulas

Coupled Differential Equations:

\frac{df}{dt} = af + br, \quad \frac{dr}{dt} = cf + dr

General Solution for Second-Order Linear Equations:

y = Ae^{m_1t} + Be^{m_2t}

Long-Term Behavior: Analyze terms $e^{mt}$ , where $m$ determines exponential growth, decay, or oscillation.

Coupled First Order Linear Equations Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.2.2 Coupled First Order Linear Equations

Introduction to Coupled Differential Equations

Worked Example: Predator-Prey Dynamics

Problem

Step 1: Rewrite as a Second-Order Equation

Step 2: Solve the Second-Order Equation

Step 3: Use Initial Conditions to Solve for Constants

Step 4: Find $r(t)$

Step 5: Long-Term Behavior

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Coupled First Order Linear Equations For their A-Level Exams.

Other Revision Notes related to Coupled First Order Linear Equations you should explore

Coupled First Order Linear Equations Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.2.2 Coupled First Order Linear Equations

Introduction to Coupled Differential Equations

Worked Example: Predator-Prey Dynamics

Problem

Step 1: Rewrite as a Second-Order Equation

Step 2: Solve the Second-Order Equation

Step 3: Use Initial Conditions to Solve for Constants

Step 4: Find r(t)r(t)r(t)

Step 5: Long-Term Behavior

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Coupled First Order Linear Equations For their A-Level Exams.

Other Revision Notes related to Coupled First Order Linear Equations you should explore

Step 4: Find $r(t)$