First order linear differential equations using an integrating factor, second order equations with constant coefficients including the complementary function and particular integral, and modelling with damped and forced systems.

What this dot point is asking

AQA wants you to solve first order linear differential equations using an integrating factor, solve second order linear equations with constant coefficients by finding the complementary function and a particular integral, apply boundary conditions, and interpret solutions of damped and forced systems such as oscillations.

First order linear equations

Second order equations: the complementary function

For the homogeneous equation $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$, try $y = e^{mx}$. Substituting gives $(am^2 + bm + c)e^{mx} = 0$, and since $e^{mx}$ is never zero this forces the auxiliary equation $am^2 + bm + c = 0$. The nature of its roots, governed by the discriminant $b^2 - 4ac$, determines the complementary function:

• Distinct real roots $m_1, m_2$ (positive discriminant): $y = Ae^{m_1 x} + Be^{m_2 x}$.
• Repeated root $m$ (zero discriminant): $y = (A + Bx)e^{m x}$. The extra factor of $x$ supplies the second independent solution, which a single exponential cannot.
• Complex roots $p \pm qi$ (negative discriminant): $y = e^{px}(A\cos qx + B\sin qx)$. The real part $p$ controls growth or decay, and $q$ sets the angular frequency of the oscillation.

The two arbitrary constants $A$ and $B$ are exactly what a second order equation needs, fixed later by two boundary or initial conditions.

The particular integral

For a non-zero right hand side $f(x)$, you add a particular integral (PI): any single solution of the full equation. You guess a trial form resembling $f(x)$ with undetermined coefficients, then substitute and match. Standard trial forms are a constant for a constant $f(x)$, a polynomial of the same degree for a polynomial, $\lambda e^{kx}$ for an exponential, and $\lambda\cos\omega x + \mu\sin\omega x$ for a trigonometric forcing term. If the trial form already appears in the complementary function, multiply it by $x$ so it is independent. The general solution is then the complementary function plus the particular integral, $y = y_{\text{CF}} + y_{\text{PI}}$, and only after writing this full solution do you apply boundary conditions to find $A$ and $B$.

Modelling damped and forced systems

The same algebra describes physical oscillators. In $\frac{d^2x}{dt^2} + k\frac{dx}{dt} + \omega^2 x = 0$ the damping term $k\frac{dx}{dt}$ sets the discriminant: light damping ($k$ small) gives complex roots and a decaying oscillation, critical damping gives a repeated root and the quickest non-oscillating return to rest, and heavy damping gives distinct real roots and a slow crawl back. Adding a periodic forcing term on the right hand side produces a particular integral that represents the steady-state response, while the complementary function (the transient) dies away when the real parts of the roots are negative. Examiners expect you to link the algebraic case directly to this physical behaviour.