Differential equations

Determine the general solution of the differential equation

giving your answer in the form y = f (x)

Given that y 5 and 12 when x = 0

determine the particular solution of the differential equation.

A rambler starts a walk at the bottom of a hill at 10 am.

The rambler walks all the way to the top of the hill and then turns around and walks back down to the bottom of the hill.

The differential equation

where A is a constant, is used to model the vertical displacement, x metres, of the rambler from the bottom of the hill, t hours after the start of the walk.

Given that after 1 hour

• the rambler has a vertical displacement of 213 m

determine the value of A to 3 significant figures.

Hence, determine the particular solution of the differential equation, giving your answer in the form x = f (t)

Use the model to determine the time at which the rambler will return to the bottom of the hill.

Given that the vertical displacement of the top of the hill is 300.68 m

use the model to find the value of t when the rambler reaches the top of the hill.

Using your answers to parts (c) and (d), give a limitation of the model.

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

Determine the general solution of the differential equation

giving your answer in the form y = f(x)

Given that y = 3 when x = 0

determine the smallest positive value of x for which y = 0

Determine the general solution of the differential equation

giving your answer in the form y = f(x)

Given that y = 3 when x = 0

determine the smallest positive value of x for which y = 0

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.

The velocity vms^–1, of a raindrop, t seconds after it falls from a cloud, is modelled by the differential equation

Initially the raindrop is at rest.

velocity of the raindrop 1 second after it falls from the cloud.

Given that the initial acceleration of the raindrop is found to be smaller than is suggested by the current model,

refine the model by changing the value of one constant.

A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge.

At time t seconds after the tourist reaches their lowest point, their vertical displacement is x metres above a fixed point 30 metres vertically above the river.

When t = 0

• x = –20

• the velocity of the tourist is 0 m s^–1

• the acceleration of the tourist is 13.6 m s^–2

In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation

where k is a positive constant.

Determine the value of k

Determine the particular solution to the differential equation.

Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point.

Give a limitation of the model.

A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge.

At time t seconds after the tourist reaches their lowest point, their vertical displacement is x metres above a fixed point 30 metres vertically above the river.

When t = 0

• x = –20

• the velocity of the tourist is 0 m s^–1

• the acceleration of the tourist is 13.6 m s^–2

In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation

where k is a positive constant.

Determine the value of k

Determine the particular solution to the differential equation.

Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point.

Give a limitation of the model.

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 community is concerned about the rising level of pollutant in its local pond and applies a chemical treatment to stop the increase of pollutant.

The concentration, x parts per million (ppm), of the pollutant in the pond water t days after the chemical treatment was applied, is modelled by the differential equation

When the chemical treatment was applied the concentration of pollutant was 3 ppm.

Use the iteration formula

once to estimate the concentration of the pollutant in the pond water 6 hours after the chemical treatment was applied.

Show that the transformation u = x³ transforms the differential equation (I) into the differential equation

Determine the general solution of equation (II)

Hence find an equation for the concentration of pollutant in the pond water t days after the chemical treatment was applied.

Find the percentage error of the estimate found in part (a) compared to the value predicted by the model, stating if it is an overestimate or an underestimate.

Two compounds, X and Y, are involved in a chemical reaction. The amounts in grams of these compounds, t minutes after the reaction starts, are x and y respectively and are modelled by the differential equations

Show that

Find, according to the model, a general solution for the amount in grams of compound X present at time t minutes.

Find, according to the model, a general solution for the amount in grams of compound Y present at time t minutes.

Given that x = 2 and y = 5 when t = 0

find

the particular solution for x,

the particular solution for y.

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

Explain whether this is supported by the model.

Two compounds, X and Y, are involved in a chemical reaction. The amounts in grams of these compounds, t minutes after the reaction starts, are x and y respectively and are modelled by the differential equations

Show that

Find, according to the model, a general solution for the amount in grams of compound X present at time t minutes.

Find, according to the model, a general solution for the amount in grams of compound Y present at time t minutes.

Given that x = 2 and y = 5 when t = 0

find

the particular solution for x,

the particular solution for y.

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

Explain whether this is supported by 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 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.

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.

A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish. At time t years, the number of white-clawed crayfish, w, and the number of signal crayfish, s, are modelled by the differential equations

Show that

Find a general solution for the number of white-clawed crayfish at time t years.

Find a general solution for the number of signal crayfish at time t years.

The model predicts that, at time T years, the population of white-clawed crayfish will have died out.

Given that w 65 and s 85 when t

find the value of T, giving your answer to 3 decimal places.

Suggest a limitation of the model.