State the integrating factor for dydx + 3y = x. [1 mark]

Write the auxiliary equation for d^2ydx^2 - dydx - 6y = 0. [1 mark]

For roots m = 3i, write the general solution. [1 mark]

CCEA CCEA A2 2021-style practice: Solve the differential equation dydx + 2y = e^x, given that y = 1 when x = 0.

This is first-order linear, so use an integrating factor I = e^int 2\,dx = e^2x. Multiply through: e^2xdydx + 2e^2xy = e^2xe^x = e^3x. The left side is ddx(ye^2x): ddx(ye^2x) = e^3x. Integrate: ye^2x = 13e^3x + C, so y = 13e^x + Ce^-2x. Apply y = 1 at x = 0: 1 = 13 + C, so C = 23. y = 13e^x + 23e^-2x. Markers reward the integrating factor, recognising the product derivative, integrating, and using the condition to find C.

Northern IrelandFurther MathsSyllabus dot point

How do you solve first-order and second-order linear differential equations?

Solving first-order linear differential equations by the integrating factor, second-order linear equations with constant coefficients via the auxiliary equation, complementary function and particular integral, and modelling with differential equations.

A CCEA A2 Further Maths answer on solving first-order linear differential equations with an integrating factor, second-order linear equations with constant coefficients using the auxiliary equation, the complementary function and particular integral, and applying them to model real situations.

Generated by Claude Opus 4.813 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

CCEA wants you to solve first-order linear differential equations using an integrating factor, solve second-order linear equations with constant coefficients via the auxiliary equation (finding the complementary function and a particular integral), and model real situations such as growth, cooling and oscillations with differential equations.

The answer

First-order linear equations

Second-order linear equations: the auxiliary equation

Particular integral and general solution

Modelling

Examples in context

Example 1. Newton's law of cooling. A hot drink cooling in a room obeys $\frac{d\theta}{dt} = -k(\theta - \theta_0)$ , a first-order linear equation. Solving it gives the exponential approach to room temperature that lets you predict when the drink is safe to hold.

Example 2. A mass on a spring with damping. The displacement of a damped oscillator satisfies a second-order linear equation, and the roots of its auxiliary equation decide whether the motion is under-damped (oscillating), critically damped or over-damped. The maths directly classifies the physical behaviour.

Try this

Q1. State the integrating factor for $\dfrac{dy}{dx} + 3y = x$ . [1 mark]

Cue. $I = e^{\int 3\,dx} = e^{3x}$ .

Q2. Write the auxiliary equation for $\dfrac{d^2y}{dx^2} - \dfrac{dy}{dx} - 6y = 0$ . [1 mark]

Cue. $m^2 - m - 6 = 0$ , so $m = 3$ or $m = -2$ .

Q3. For roots $m = \pm 3i$ , write the general solution. [1 mark]

Cue. $y = A\cos 3x + B\sin 3x$ .

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA A2 20217 marksSolve the differential equation

\dfrac{dy}{dx} + 2y = e^{x}

, given that

y = 1

when

x = 0

Show worked answer →

This is first-order linear, so use an integrating factor $I = e^{\int 2\,dx} = e^{2x}$ .

Multiply through: $e^{2x}\dfrac{dy}{dx} + 2e^{2x}y = e^{2x}e^{x} = e^{3x}$ . The left side is $\dfrac{d}{dx}\left(ye^{2x}\right)$ :

$\dfrac{d}{dx}\left(ye^{2x}\right) = e^{3x}.$

Integrate: $ye^{2x} = \dfrac{1}{3}e^{3x} + C$ , so $y = \dfrac{1}{3}e^{x} + Ce^{-2x}$ .

Apply $y = 1$ at $x = 0$ : $1 = \dfrac{1}{3} + C$ , so $C = \dfrac{2}{3}$ .

$y = \dfrac{1}{3}e^{x} + \dfrac{2}{3}e^{-2x}.$

Markers reward the integrating factor, recognising the product derivative, integrating, and using the condition to find $C$ .

CCEA A2 20198 marksSolve

\dfrac{d^2y}{dx^2} - 5\dfrac{dy}{dx} + 6y = 12

, finding the general solution.

Show worked answer →

The auxiliary equation is $m^2 - 5m + 6 = 0$ , which factorises as $(m - 2)(m - 3) = 0$ , so $m = 2$ or $m = 3$ .

The complementary function is $y_c = Ae^{2x} + Be^{3x}$ .

For the particular integral, the right-hand side is a constant, so try $y_p = k$ (a constant). Then $y_p'' = y_p' = 0$ , and substituting gives $6k = 12$ , so $k = 2$ .

The general solution is the sum:

$y = Ae^{2x} + Be^{3x} + 2.$

Markers reward the auxiliary equation, the complementary function from its roots, a constant trial for the particular integral, and the combined general solution.

Related dot points

Sources & how we know this

CCEA GCE Further Mathematics specification — CCEA (2018)