A-level · Integration

Differential equationsExam-style questions

Solving first-order separable differential equations, using a cooling block of metal throughout. Writing a differential equation from a description of a rate; separating the variables; integrating both sides with one constant; turning a log into an explicit formula; using the conditions to find the constants; and reading the model — its long-term behaviour and its limits, including a logistic model that needs partial fractions.

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57 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Express $\dfrac{3}{x(6 - x)}$ in partial fractions.

(2 marks)

A disease is spreading through a herd of $600$ cattle on a farm. The number of cattle that have caught the disease, $x$ hundred, $t$ weeks after the disease was first found in the herd, is modelled by the differential equation

$\dfrac{dx}{dt} = \dfrac{x(6 - x)}{3(t + 1)}$

When the disease was first found, $100$ of the cattle had caught it.
(b) Solve the differential equation to show that, according to the model,

$x = \dfrac{6(t + 1)^2}{(t + 1)^2 + a}$

where $a$ is a constant to be found.

(6 marks)
(c) Find, according to the model, how long it takes for $400$ of the cattle to have caught the disease. Give your answer in weeks to 3 significant figures.

(2 marks)
Level 5 · 10 marksTry it
2
The number of tickets $N$ sold for a concert, $t$ days after going on sale, is modelled by $\dfrac{dN}{dt} = \dfrac{1}{16000}N(8000 - N)$ , with $N = 1000$ when $t = 0$ .
(a) Solve the differential equation, giving $N$ in terms of $t$ .
(b) State the capacity of the venue, assuming it sells out as $t \to \infty$ .
Level 5 · 7 marksTry it
3
A population of fish $P$ in a lake, $t$ years after stocking, is modelled by $\dfrac{dP}{dt} = \dfrac{1}{2500}P(500 - P)$ , with $P = 50$ when $t = 0$ .
(a) Show that $P = \dfrac{500}{1 + 9e^{-t/5}}$ .
(b) State the long-term population of the lake.
Level 5 · 7 marksTry it
4
A hot liquid cools so that its temperature $\theta\,^\circ\text{C}$ at time $t$ minutes satisfies $\dfrac{\mathrm{d}\theta}{\mathrm{d}t} = -k(\theta - 20)$ , where $k$ is a positive constant.
(a) Show that $\theta = 20 + A e^{-kt}$ , where $A$ is a constant. (3 marks)
(b) Initially $\theta = 80$ , and after $10$ minutes $\theta = 50$ . Find the values of $A$ and $k$ . (4 marks)
(c) Find the temperature after $20$ minutes, and state the temperature the liquid approaches in the long term. (3 marks)
Level 5 · 10 marksTry it
5
A radioactive substance decays so that its mass $m$ grams at time $t$ years satisfies $\dfrac{\mathrm{d}m}{\mathrm{d}t} = -km$ .
(a) Show that $m = m_0 e^{-kt}$ , where $m_0$ is the initial mass. (2 marks)
(b) The half-life of the substance is $5$ years. Show that $k = \tfrac15\ln 2$ . (3 marks)
(c) Find, to the nearest year, the time for the substance to decay to $10\%$ of its initial mass. (4 marks)
Level 5 · 9 marksTry it

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