Differential equations

Water flows at a constant rate into a large container. There is a tap at the bottom of the container. At time t hours after the tap was opened

• the volume of water in the container is Vm³

• water is flowing into the container at a constant rate of 0.45 m³ per hour

• water is leaving the container through the tap at a rate of 0.3Vm³ per hour

Show that

Given that when the tap was opened, there was 0.25 m³ of water in the container,

solve the differential equation to show that

where P, Q and k are positive constants to be found. Given that

• the capacity of the container is 2 m³

• the tap remains open

• the water continues to flow into the tank at the same rate

determine whether the container will ever become full, giving a reason for your answer.

Figure 2 shows a cylindrical tank of height 1.5 m.

Initially the tank is full of water.

The water starts to leak from a small hole, at a point L, in the side of the tank.

While the tank is leaking, the depth, H metres, of the water in the tank is modelled by the differential equation

where t hours is the time after the leak starts.

Using the model,

show that

where A and B are constants to be found,

find the time taken for the depth of the water to decrease to 1.2 m. Give your answer in hours and minutes, to the nearest minute.

In the long term, the water level in the tank falls to the same height as the hole.

Find, according to the model, the height of the hole from the bottom of the tank.

A balloon is being inflated.

In a simple model,

• the balloon is modelled as a sphere

• the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon

At time t seconds, the radius of the balloon is rcm.

Write down a differential equation to model this situation.

At the instant when t = 10

• the radius is 16 cm

• the radius is increasing at a rate of 0.9 cm s^–1

Solve the differential equation to show that

Hence find the radius of the balloon when t = 20

Give your answer to the nearest millimetre.

Suggest a limitation of the model.

The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation

The plant cell has an initial volume of 20 microlitres.

Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.

Show that

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

The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell.

Find the value of L, giving a reason for your answer.

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, Vm³, t hours after the chemicals were mixed, is modelled by the differential equation

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

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

the time delay giving your answer in minutes,

the limit giving your answer in m³

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 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 0.1hm³ per minute

Show that, according to the model,

Given that when the tap was opened, the depth of water in the tank was 2 m,

show that, according to the model,

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.

A large spherical balloon is deflating. At time t seconds the balloon has radius rcm and volume Vcm³ The volume of the balloon is modelled as decreasing at a constant rate.

Using this model, show that

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 the 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.

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

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.

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

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,

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.

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.