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Differential Equations · Paper 3 — Free Maths AA HL Practice Questions

What this topic covers

Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.

Read the full Calculus lesson notes →

1ChallengeLAQSolving first-order differential equationss10 marksPaper 3~15 min

A population of a bird species on an isolated island has size

P

(in hundreds) at time

t

(in years), satisfying the logistic differential equation

\frac{dP}{dt} = \frac{1}{5}P\!\left(1 - \frac{P}{12}\right), \quad t \geq 0,

with initial condition

P(0) = 3

(a)

By separating variables and using partial fractions, show that

P(t) = \frac{12}{1 + 3e^{-t/5}}.

[5 marks]

(b)

Find the limiting value of

P(t)

t \to \infty

and state what this represents in context. [2 marks]

(c)

The differential equation has two equilibrium solutions. Justify, by analysing the sign of

\dfrac{dP}{dt}

in each region, that the equilibrium

P = 12

is stable and

P = 0

is unstable. Hence explain why the result in (b) holds for any initial condition

P(0) = P_0

where

0 < P_0 < 12

, but not for

P_0 = 0

. [3 marks]

Solutions

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