Forming differential equations

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

Two different colours of paint are being mixed together in a container.

The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container.

Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint.

The colour of the paint mixture is now altered by

• adding red paint to the container at a rate of 2 litres per second

• adding blue paint to the container at a rate of 1 litre per second

• pumping fully mixed paint from the container at a rate of 3 litres per second.

Let r litres be the amount of red paint in the container at time t seconds after the colour of the paint mixture starts to be altered.

Show that the amount of red paint in the container can be modelled by the differential equation

where α is a positive constant to be determined.

By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second.

It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.

Use this information to evaluate the model, giving a reason for your answer.

Two different colours of paint are being mixed together in a container.

The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container.

Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint.

The colour of the paint mixture is now altered by

• adding red paint to the container at a rate of 2 litres per second

• adding blue paint to the container at a rate of 1 litre per second

• pumping fully mixed paint from the container at a rate of 3 litres per second.

Let r litres be the amount of red paint in the container at time t seconds after the colour of the paint mixture starts to be altered.

Show that the amount of red paint in the container can be modelled by the differential equation

where α is a positive constant to be determined.

By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second.

It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.

Use this information to evaluate the model, giving a reason for your answer.

A sample of bacteria in a sealed container is being studied.

The number of bacteria, P, in thousands, is modelled by the differential equation

where t is the time in hours after the start of the study.

Initially, there are exactly 5000 bacteria in the container.

Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.

Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.

State a limitation of the model.

A sample of bacteria in a sealed container is being studied.

The number of bacteria, P, in thousands, is modelled by the differential equation

where t is the time in hours after the start of the study.

Initially, there are exactly 5000 bacteria in the container.

Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.

Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.

State a limitation of the model.

A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination. The concentration of antibodies, x, measured in micrograms (μg) per millilitre (ml) of blood, is modelled by the differential equation

where t is the number of weeks since the vaccination was given.

Find a general solution of the differential equation.

Initially, • there are no antibodies in the bloodstream of the patient • the concentration of antibodies is estimated to be increasing at 10 μg/ml per week

Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination.

A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than 5 μg/ml.

Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose.

A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination. The concentration of antibodies, x, measured in micrograms (μg) per millilitre (ml) of blood, is modelled by the differential equation

where t is the number of weeks since the vaccination was given.

Find a general solution of the differential equation.

Initially, • there are no antibodies in the bloodstream of the patient • the concentration of antibodies is estimated to be increasing at 10 μg/ml per week

Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination.

A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than 5 μg/ml.

Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose.

Show that v=ab^x, where a and b are constants to be found.

The time taken for the speed of P to decrease from 4 m s⁻¹ to 2 m s⁻¹ is T seconds.

A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.

Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt.

It is assumed that the salt water mixes instantly with the contents of the tank upon entry.

At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute.

Given that there are S grams of salt in the tank after t minutes,

=show that the situation can be modelled by the differential equation

Hence find the number of grams of salt in the tank after 10 minutes.

When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.

=Find, to the nearest minute, when the valve would need to be closed.

Evaluate the model.

A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.

Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt.

It is assumed that the salt water mixes instantly with the contents of the tank upon entry.

At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute.

Given that there are S grams of salt in the tank after t minutes,

show that the situation can be modelled by the differential equation

Hence find the number of grams of salt in the tank after 10 minutes.

When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.

Find, to the nearest minute, when the valve would need to be closed.

Evaluate the model.