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Differential Equations (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

2 hours18 questions
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Medium
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Very Hard
14 marks

Find the general solution to the differential equation

fraction numerator straight d y over denominator straight d x end fraction equals 2 x y

where y greater than 0.

How did you do?

24 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 3 x squared y

where y greater than 0.

How did you do?

3a1 mark

The differential equation

fraction numerator straight d V over denominator straight d t space end fraction equals negative k V

is used to model the rate at which water is leaking from a container, where

  • V is the volume of water in the container

  • t is the time in seconds

  • k is a positive constant

Explain, in context, the significance of the negative sign in the model.

How did you do?

3b3 marks

Find the general solution to the differential equation.

How did you do?

3c2 marks

Given that

  • k equals 0.02

  • the initial volume of the container is 300 litres

find a complete equation linking V and t.

How did you do?

4a4 marks

Given that y greater than 1, find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 6 x squared open parentheses y minus 1 close parentheses

writing your answer in the form

y equals A straight e to the power of straight f left parenthesis x right parenthesis end exponent plus 1

where A is a constant and straight f open parentheses x close parentheses is a function of x which you should find.

How did you do?

4b4 marks

Given that y greater than negative 2, find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 9 left parenthesis y plus 2 right parenthesis square root of x

writing your answer in the form

y equals A straight e to the power of straight f open parentheses x close parentheses end exponent minus 2

where A is a constant and straight f open parentheses x close parentheses is a function of x which you should find.

How did you do?

5a2 marks

The volume of water in a sink, V, decreases with time t, measured from the point at which the plug is removed.

It is known that Vdecreases at a rate proportional to its volume.

Use this information to write down a suitable differential equation for V and t, using a constant of proportionality k where k greater than 0.

How did you do?

5b2 marks

The general solution to the differential equation in part (a) can be written in the form

V equals A straight e to the power of negative k t end exponent

where k greater than 0.

(i) State, in the context of the question, what the constant A represents.

(ii) Briefly explain the significance of the negative sign in the solution.

How did you do?

6a3 marks

A differential equation is given by

fraction numerator d y over denominator d x end fraction equals sec squared x

where y equals 2 square root of 3 when x equals pi over 3.

Show that

y equals a plus tan space x

where a is a constant to be found.

How did you do?

6b5 marks

A differential equation is given by

sec space x fraction numerator d y over denominator d x end fraction equals cosec space y

where y equals 0 when x equals pi over 2.

Show that

cos space y equals b minus sin space x

where b is a constant to be found.

How did you do?

1
Sme Calculator4 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals sin squared open parentheses 2 y close parentheses

giving your answer in the form x equals straight f left parenthesis y right parenthesis plus c where c is a constant and straight f open parentheses y close parentheses is a function to be found.

How did you do?

2a2 marks

Find the general solution to the differential equation

9 t squared minus 4 plus fraction numerator d x over denominator d t end fraction equals 0

How did you do?

2b3 marks

Find the particular solution to the differential equation

fraction numerator d V over denominator d x end fraction minus 4 equals 2 straight e to the power of x

given that the graph of V against x passes through the point with coordinates open parentheses 0 comma space 3 close parentheses.

How did you do?

3a5 marks

A differential equation is given by

straight e to the power of negative 3 x end exponent fraction numerator d y over denominator d x end fraction equals 2 straight e to the power of y

It is known that y equals 0 when x equals 0.

Solve the differential equation, giving your answer in the form

p straight e to the power of 3 x end exponent plus straight e to the power of negative y end exponent equals q

where p and q are rational numbers to be found.

How did you do?

3b6 marks

A differential equation is given by

sin squared x fraction numerator d y over denominator d x end fraction equals cos squared y

where y equals 0 when x equals pi over 4.

Solve the differential equation, giving your answer in the form

tan space y equals straight f open parentheses x close parentheses

where straight f open parentheses x close parentheses is a function to be found.

How did you do?

4a
Sme Calculator3 marks

A weather balloon of volume V m3 is being inflated, where t is the time in minutes after inflation begins.

  • The rate of change of its volume is inversely proportional to its volume

  • When the rate of inflation of the balloon is 10 m3 min-1, the volume of the balloon is 20 m3

Use this information to write down a suitable differential equation for V and t.

How did you do?

4b
Sme Calculator3 marks

Show that the general solution to the differential equation is

V squared equals 400 t plus c

where c is a constant.

How did you do?

4c
Sme Calculator3 marks

Initially, the balloon is flat with a volume of 0 m3.

Find the volume of the balloon after 25 minutes.

How did you do?

5a7 marks

A disease affecting trees is spreading throughout a large forested area. Let N be the number of infected trees t days after the disease was first discovered.

A model for N and t is given by

space fraction numerator d N over denominator d t end fraction equals k N t

where k is a positive constant.

It is known that

  • When the disease was first discovered, 3 trees were infected

  • Ten days after the disease was first discovered, 10 trees were infected

Solve the differential equation to show that

N equals 3 straight e to the power of a t squared end exponent

where

a equals 1 over 100 ln open parentheses 10 over 3 close parentheses

How did you do?

5b
Sme Calculator3 marks

Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.

Find the number of days (since first discovering the disease) that the model predicts scientists have in order to take action.

How did you do?

1a4 marks

Find the general solution to the differential equation

fraction numerator 2 y minus 1 over denominator 3 end fraction fraction numerator d y over denominator d x end fraction equals x squared y squared minus x squared y

where y greater than 1, giving your answer in the form

y squared minus y equals straight f open parentheses x close parentheses

How did you do?

1b4 marks

Find the general solution to the differential equation

3 fraction numerator d y over denominator d x end fraction equals fraction numerator cosec space y cubed over denominator space y squared end fraction

giving your answer in the form x equals straight g left parenthesis y right parenthesis.

How did you do?

2a2 marks

A hot air balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m3) and time (seconds), write down a differential equation to describe the relationship between volume and time as the hot air balloon is inflated.

How did you do?

2b
Sme Calculator8 marks

You are given the following information:

  • Initially, the hot air balloon has a volume of zero

  • After 400 seconds of inflating, its volume is 600 m3

  • The hot air balloon is considered ready for release when its volume reaches 1250 m3

If the hot air balloon needs to be ready for release by midday, find the latest time that it can start being inflated.

How did you do?

3a2 marks

Find the general solution to the differential equation

1 half sec squared open parentheses 3 t close parentheses plus 2 fraction numerator d x over denominator d t end fraction equals 0

How did you do?

3b6 marks

Find the particular solution to the differential equation

2 x straight e to the power of 4 x end exponent minus 3 fraction numerator d V over denominator d x end fraction equals 1

where the graph of V against x passes through the point with coordinates open parentheses 0 comma space 2 close parentheses.

How did you do?

15 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 2 x y plus 2 x minus y minus 1

where y greater than negative 1, giving your answer in the form y equals straight f open parentheses x close parentheses.

How did you do?

2a6 marks

Palm trees are being planted on an island. Let N be the total number of palm trees planted on the island after t days.

The variables N and t are modelled by the differential equation

fraction numerator straight d N over denominator straight d t end fraction equals k N open parentheses N minus 1 close parentheses comma space space space N greater than 1

where N greater than 1 and k is a positive constant.

By solving the differential equation, show that

N equals fraction numerator 1 over denominator 1 minus A straight e to the power of k t end exponent end fraction

where A is a positive constant.

How did you do?

2b3 marks

It is known that

  • Initially 2 palm trees are planted

  • After 14 days, 4 palm trees in total have been planted

Use this information to show that

k equals 1 over 14 ln space p

where p is a rational number to be found.

How did you do?

2c3 marks

By considering the form of the solution to the differential equation, suggest a range of values of t for which the model is valid.

How did you do?

3a6 marks

The temperature of a heated object, T°C, cools over time, t minutes. The room temperature (called the ambient temperature) is constant, T subscript amb, where T greater than T subscript amb.

Newton’s Law of Cooling states that the rate of decrease in temperature of a heated object is directly proportional to the difference between the object’s temperature and the ambient temperature.

By forming and solving a differential equation in T and t (involving the constant T subscript amb and a positive constant of proportionality, k) show that

T equals T subscript amb plus A straight e to the power of negative k t end exponent

How did you do?

3b
Sme Calculator4 marks

For food safety reasons, a meat processing factory must store its products at a temperature of below -1 °C.

  • One particular product has a temperature of 7 °C

  • It is placed in one of the factory's freezers, which has a constant ambient temperature of -4 °C

  • One minute later, its temperature has dropped to 4.7 °C.

  • Any products that fail to cool to below -1 °C within 6 minutes must be discarded

Determine whether or not this product will need to be discarded.

How did you do?

4a4 marks

Show that the solution to the differential equation

cos x fraction numerator d y over denominator d x end fraction equals cos space y space

where y equals pi when x equals 0 may be written in the form

open vertical bar tan space open parentheses y over 2 plus pi over 4 close parentheses space close vertical bar equals open vertical bar tan space open parentheses x over 2 plus pi over 4 close parentheses close vertical bar

How did you do?

4b
Sme Calculator6 marks

(i) Prove that if open vertical bar tan space open parentheses y over 2 plus pi over 4 close parentheses close vertical bar equals open vertical bar tan open parentheses x over 2 plus pi over 4 close parentheses close vertical bar then

y equals x plus 2 n pi space space space space space space or space space space space space space y equals negative x plus open parentheses 2 n minus 1 close parentheses pi

where n is an integer.

(ii) Hence deduce that the particular solution to the differential equation in part (a) is

y equals pi minus x

How did you do?