First order linear differential equations by the integrating factor, second order linear constant-coefficient equations via the auxiliary equation and particular integral, and applications to simple harmonic motion and damped systems.

What this dot point is asking

OCR wants you to solve first order linear differential equations using an integrating factor, solve second order linear constant-coefficient equations by forming the auxiliary equation for the complementary function and adding a particular integral for a non-zero right-hand side, and interpret second order equations as models of simple harmonic motion and of damped systems, classifying the damping from the nature of the roots.

First order linear equations: the integrating factor

A first order equation is linear when it can be written $\dfrac{dy}{dx} + P(x)y = Q(x)$. Multiplying by the integrating factor turns the left-hand side into the derivative of a product, which you can then integrate directly.

Second order equations: complementary function

A homogeneous second order equation $a\ddot{x} + b\dot{x} + cx = 0$ is solved through its auxiliary equation $am^2 + bm + c = 0$. The form of the solution depends on the roots.

Particular integral and the general solution

When the right-hand side is non-zero, $a\ddot{x} + b\dot{x} + cx = f(t)$, add a particular integral (a trial function shaped like $f$, with coefficients fixed by substitution) to the complementary function. The full general solution is the sum, and any initial conditions then determine $A$ and $B$.

Simple harmonic motion and damping

The equation $\ddot{x} + \omega^2 x = 0$ has auxiliary roots $\pm \omega i$, giving $x = A\cos\omega t + B\sin\omega t$: simple harmonic motion of angular frequency $\omega$. Adding a resistance term $b\dot{x}$ models damping, and the auxiliary roots classify it.

• Light (under) damping. Complex roots $p \pm qi$ with $p < 0$: the system oscillates with exponentially decaying amplitude.
• Critical damping. A repeated real root: the fastest return to rest without oscillating.
• Heavy (over) damping. Two distinct negative real roots: a slow return to rest with no oscillation.

Differential equations draw on integration, on complex numbers (for the oscillatory case), and connect Pure Core to the mechanics of oscillating and damped systems.

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 general solution of $\ddot{x} + 9x = 0$. [2 marks]

• Cue. Auxiliary roots $\pm 3i$, so $x = A\cos 3t + B\sin 3t$ (SHM with $\omega = 3$).