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

A large spherical balloon is deflating.

At time $t$ seconds the balloon has radius $r$ cm and volume $V$ cm³.

The volume of the balloon is modelled as decreasing at a constant rate.

Using this model, show that

$\frac{\mathrm{d}r}{\mathrm{d}t}=-\frac{k}{r^2}$

where $k$ is a positive constant.

Given that

• the initial radius of the balloon is 40 cm

• after 5 seconds the radius of the balloon is 20 cm

• the volume of the balloon continues to decrease at a constant rate until the balloon is empty

solve a differential equation to find a complete equation linking $r$ and $t$.

Find the limitation on the values of $t$ for which the equation in part (b) is valid.

Find the general solution to the differential equation

$\frac{dy}{dx}={\sin }^2\left(2y\right)$

giving your answer in the form $x=\mathrm{f}\left(y\right)+c$ where $c$ is a constant and $\mathrm{f}\left(y\right)$ is a function to be found.

Find the general solution to the differential equation

$9t^2-4+\frac{dx}{dt}=0$

Find the particular solution to the differential equation

$\frac{dV}{dx}-4=2{\mathrm{e}}^x$

given that the graph of $V$ against $x$ passes through the point with coordinates $\left(0,3\right)$.

A differential equation is given by

${\mathrm{e}}^{-3x}\frac{dy}{dx}=2{\mathrm{e}}^y$

It is known that $y=0$ when $x=0$.

Solve the differential equation, giving your answer in the form

$p{\mathrm{e}}^{3x}+{\mathrm{e}}^{-y}=q$

where $p$ and $q$ are rational numbers to be found.

A differential equation is given by

${\sin }^{2}x\frac{dy}{dx}={\cos }^{2}y$

where $y=0$ when $x=\frac{\pi }{4}$.

Solve the differential equation, giving your answer in the form

$\tan y=\mathrm{f}\left(x\right)$

where $\mathrm{f}\left(x\right)$ is a function to be found.

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 $t$.

Show that the general solution to the differential equation is

$V^2=400t+c$

where $c$ is a constant.

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

Find the volume of the balloon after 25 minutes.

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

$\frac{dN}{dt}=kNt$

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

$N=3{\mathrm{e}}^{a{t}^2}$

where

$a=\frac{1}{100}\ln \left(\frac{10}{3}\right)$

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.

Find the general solution to the differential equation

$\frac{1}{2}{\sec }^2\left(3t\right)+2\frac{dx}{dt}=0$

Find the particular solution to the differential equation

$2x{\mathrm{e}}^{4x}-3\frac{dV}{dx}=1$

where the graph of $V$ against $x$ passes through the point with coordinates $\left(0,2\right)$.

The height above ground, $H$ metres, of a passenger on a roller coaster can be modelled by the differential equation

$\frac{\mathrm{d}H}{\mathrm{d}t}=\frac{H\cos \left(0.25t\right)}{40}$

where $t$ is the time, in seconds, from the start of the ride.

Given that the passenger is 5 m above the ground at the start of the ride,

show that $H=5{\mathrm{e}}^{0.1\sin \left(0.25t\right)}$.

State the maximum height of the passenger above the ground.

The passenger reaches the maximum height, for the second time, $T$ seconds after the start of the ride.

Find the value of $T$.

Express $\frac{3}{\left(2x-1\right)\left(x+1\right)}$ in partial fractions.

When chemical $A$ and chemical $B$ are mixed, oxygen is produced.

A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time.

The total volume of oxygen produced, $V$ m³, $t$ hours after the chemicals were mixed, is modelled by the differential equation

$\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{3V}{\left(2t-1\right)\left(t+1\right)}V\ge 0t\ge k$

where $k$ is a constant.

Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m³ of oxygen had been produced, solve the differential equation to show that

$V=\frac{3\left(2t-1\right)}{\left(t+1\right)}$

The scientist noticed that

• there was a time delay between the chemicals being mixed and oxygen being produced

• there was a limit to the total volume of oxygen produced

Deduce from the model

(i) the time delay giving your answer in minutes,

(ii) the limit giving your answer in m³

A tank in the shape of a cuboid is being filled with water.

The base of the tank measures 20 m by 10 m and the height of the tank is 5 m, as shown in Figure 1.

At time $t$ minutes after water started flowing into the tank the height of the water was $h$ m and the volume of the water in the tank was $V$ m³.

In a model of this situation

• the sides of the tank have negligible thickness

• the rate of change of $V$ is inversely proportional to the square root of $h$

Show that

$\frac{dh}{dt}=\frac{\lambda }{\sqrt{h}}$

where $\lambda$ is a constant.

Given that

• initially the height of the water in the tank was 1.44 m

• exactly 8 minutes after water started flowing into the tank the height of the water was 3.24 m

use the model to find an equation linking $h$ with $t$, giving your answer in the form

$h^{\frac{3}{2}}=At+B$

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

Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full.

Find the general solution to the differential equation

$\frac{2y-1}{3}\frac{dy}{dx}=x^2{y}^2-x^{2}y$

where $y>1$, giving your answer in the form

$y^2-y=\mathrm{f}\left(x\right)$

Find the general solution to the differential equation

$3\frac{dy}{dx}=\frac{\mathrm{cosec}{y}^3}{y^2}$

giving your answer in the form $x=\mathrm{g}\left(y\right)$.

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.

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.

Water is flowing into a large tank. The depth of water, $h$ metres, at time $t$ minutes satisfies the differential equation

$\frac{\mathrm{d}h}{\mathrm{d}t}=\frac{\sqrt{h}\left(4-\sqrt{h}\right)}{8}0\le h<16$

Use the substitution $h=x^2$ to show that

$\int \frac{16}{4-x}\mathrm{d}x=t+c$

Given that the tank is initially empty, find the time taken for the depth of water to reach 9 metres.

Give your answer to the nearest minute.

Find the general solution to the differential equation

$\frac{dy}{dx}=2xy+2x-y-1$

where $y>-1$, giving your answer in the form $y=\mathrm{f}\left(x\right)$.

A spherical mint of radius 5 mm is placed in the mouth and sucked.

Four minutes later, the radius of the mint is 3 mm.

In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.

Using this model and all the information given, find an equation linking the radius of the mint and the time.

(You should define the variables that you use.)

Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.

Suggest a limitation of the model.

Use the substitution $u=4-\sqrt{h}$ to show that

$\int \frac{dh}{4-\sqrt{h}}=-8\ln \left|4-\sqrt{h}\right|-2\sqrt{h}+k$

where $k$ is a constant.

A team of scientists is studying a species of slow growing tree.

The rate of change in height of a tree in this species is modelled by the differential equation

$\frac{dh}{dt}=\frac{t^{0.25}\left(4-\sqrt{h}\right)}{20}$

where $h$ is the height in metres and $t$ is the time, measured in years, after the tree is planted.

Find, according to the model, the range in heights of trees in this species.

One of these trees is one metre high when it is first planted.

According to the model, calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.

Water flows at a constant rate into a large tank.

The tank is a cuboid, with all sides of negligible thickness.

The base of the tank measures 8 m by 3 m and the height of the tank is 5 m.

There is a tap at a point $T$ at the bottom of the tank, as shown in Figure 5.

At time $t$ minutes after the tap has been opened

• the depth of the water in the tank is $h$ metres

• water is flowing into the tank at a constant rate of $0.48$ m³ per minute

• water is modelled as leaving the tank through the tap at a rate of $0.1h$ m³ per minute

Show that, according to the model,

$1200\frac{\mathrm{d}h}{\mathrm{d}t}=24-5h$

Given that when the tap was opened, the depth of the water in the tank was 2 m, show that, according to the model,

$h=A+B{\mathrm{e}}^{-kt}$

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

Given that the tap remains open, determine, according to the model, whether the tank will ever become full, giving a reason for your answer.

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

$\frac{\mathrm{d}N}{\mathrm{d}t}=kN\left(N-1\right),N>1$

where $N>1$ and $k$ is a positive constant.

By solving the differential equation, show that

$N=\frac{1}{1-A{\mathrm{e}}^{kt}}$

where $A$ is a positive constant.

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

$k=\frac{1}{14}\ln p$

where $p$ is a rational number to be found.

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

The rate of increase of a population $P$ of microorganisms at time $t$ seconds is given by

$\frac{\mathrm{d}P}{\mathrm{d}t}=kP\left(5-P\right)$

where $k$ is a positive constant.

Given that $P=1$ when $t=0$ and $P=2$ when $t=2$, show that $P=\frac{5}{1+4e^{-5kt}}$.

Hence find the exact value of $k$.

State the limiting value of $P$ as $t\to \infty$.

The temperature of a heated object, $T$°C, cools over time, $t$ minutes. The room temperature (called the ambient temperature) is constant, $T_{\mathrm{amb}}$, where $T>T_{\mathrm{amb}}$.

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 $T_{\mathrm{amb}}$ and a positive constant of proportionality, $k$) show that

$T=T_{\mathrm{amb}}+A{\mathrm{e}}^{-kt}$

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.

Show that the solution to the differential equation

$\cos x\frac{dy}{dx}=\cos y$

where $y=\pi$ when $x=0$ may be written in the form

$\left|\tan \left(\frac{y}{2}+\frac{\pi }{4}\right)\right|=\left|\tan \left(\frac{x}{2}+\frac{\pi }{4}\right)\right|$

(i) Prove that if $\left|\tan \left(\frac{y}{2}+\frac{\pi }{4}\right)\right|=\left|\tan \left(\frac{x}{2}+\frac{\pi }{4}\right)\right|$ then

$y=x+2n\pi \mathrm{or}y=-x+\left(2n-1\right)\pi$

where $n$ is an integer.

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

$y=\pi -x$