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 tree is planted in a park.

A model for the growth of the tree is

$\frac{dH}{dt} = 0.9e^{-0.15t}$

where $H$ metres is the height of the tree $t$ years after it is planted.

When the tree is planted, its height is $1.2$ m.

Using the model,
(a) show that

$H = A + Be^{-0.15t}$

where $A$ and $B$ are constants to be found,

(3 marks)
(b) find how long after it is planted the tree is $5$ m tall. Give your answer in years and months, to the nearest month.

(3 marks)

In the long term, the tree stops growing.
(c) Use the model to find the height of the tree once it is fully grown.

(2 marks)
Level 4 · 8 marksTry it
2
A chemical is added to the water in a swimming pool.

The concentration of the chemical in the water, $C$ $C$ milligrams per litre, $t$ $t$ hours after the chemical is added, is modelled by a differential equation.

In the model, the rate of decrease of $C$ $C$ is proportional to $C$ $C$ .
(a) Write down a differential equation to model this situation.

(1 mark)

Given that
- the initial concentration of the chemical was $3$ milligrams per litre
- exactly $6$ hours after the chemical was added, the concentration was $2.4$ milligrams per litre
(b) solve the differential equation to find a complete equation for the model, giving the value of the constant to 3 significant figures.

(5 marks)
(c) State a limitation of the model.

(1 mark)
Level 4 · 7 marksTry it
3
A carton of soup is taken out of a freezer and left to stand in a kitchen.

The temperature of the kitchen is a constant $21\ ^\circ$ $21^{\circ}$ C.

In a model of this situation, the rate of increase of the temperature of the soup is proportional to the difference between the temperature of the kitchen and the temperature of the soup.

At time $t$ $t$ minutes after the carton is taken out of the freezer, the temperature of the soup is $\theta\ ^\circ$ $θ^{\circ}$ C.
(a) Write down a differential equation to model this situation.

(1 mark)

Given that
- when the carton is taken out of the freezer the temperature of the soup is $-4\ ^\circ$ C
- exactly $25$ minutes later the temperature of the soup is $6\ ^\circ$ C
(b) solve the differential equation to find a complete equation linking $\theta$ $θ$ and $t$ $t$ , giving the value of the constant of proportionality to 3 significant figures.

(5 marks)

As $t$ $t$ increases, the temperature of the soup approaches a limiting value $L\ ^\circ$ $L^{\circ}$ C.
(c) Find the value of $L$ $L$ , giving a reason for your answer.

(2 marks)
Level 4 · 8 marksTry it
4
A tank is draining. At time $t$ minutes the depth of water in the tank is $h$ metres, and the depth is modelled by the differential equation

$\dfrac{\mathrm{d}h}{\mathrm{d}t} = -k\sqrt{h}$

where $k$ is a positive constant. When $t = 0$ the depth is $4$ m.
(a) Solve the differential equation to show that $\sqrt{h} = 2 - \dfrac{kt}{2}$ . (4 marks)
(b) Given that the depth is $2.25$ m when $t = 10$ , find the value of $k$ . (2 marks)
(c) Find the depth of water in the tank when $t = 20$ . (2 marks)
(d) Find the time at which the model predicts the tank will be empty. (2 marks)
Level 4 · 10 marksTry it
5
A population of insects is modelled by the differential equation

$\dfrac{\mathrm{d}P}{\mathrm{d}t} = 0.4P\cos(0.5t)$

where $P$ is the population in thousands and $t$ is the time in weeks. When $t = 0$ , $P = 6$ .
(a) Solve the differential equation to show that

$P = 6\mathrm{e}^{0.8\sin(0.5t)}$

(6 marks)
(b) Find the maximum population predicted by this model, giving your answer to $3$ significant figures. (2 marks)
Level 4 · 8 marksTry it

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