Second order 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 scientist is studying the effect of introducing a population of type A bacteria into a population of type B bacteria.

At time t days, the number of type A bacteria, x, and the number of type B bacteria, y, are modelled by the differential equations

Show that

Determine a general solution for the number of type A bacteria at time t days.

Determine a general solution for the number of type B bacteria at time t days.

The model predicts that, at time T hours, the number of bacteria in the two populations will be equal.

Given thatx = 100 and y = 275 when t = 0

determine the value of T, giving your answer to 2 decimal places.

Suggest a limitation of the model.

A scientist is studying the effect of introducing a population of type A bacteria into a population of type B bacteria.

At time t days, the number of type A bacteria, x, and the number of type B bacteria, y, are modelled by the differential equations

Show that

Determine a general solution for the number of type A bacteria at time t days.

Determine a general solution for the number of type B bacteria at time t days.

The model predicts that, at time T hours, the number of bacteria in the two populations will be equal.

Given thatx = 100 and y = 275 when t = 0

determine the value of T, giving your answer to 2 decimal places.

Suggest a limitation of the model.

The motion of a particle P along the x-axis is modelled by the differential equation

where P has displacement x metres from the origin O at time t minutes,t 0

determine the particular solution of the differential equation (I)

The vertical height, h m, above horizontal ground, of a passenger on a fairground ride, t seconds after the ride starts, where t 5, is modelled by the differential equation

Given that t = e^x, show that

Hence show that the transformation t = e^x transforms equation (I) into the equation

Hence show that

where A and B are constants.

Given that when t = 1, h = 2.5 and when t = 2, = –1

determine the height of the passenger above the ground 5 seconds after the start of the ride.

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.

A particle P moves along a straight line.

At time t minutes, the displacement, x metres, of P from a fixed point O on the line is modelled by the differential equation

Show that the transformation x = ty transforms equation (I) into the equation

Hence find a general solution for the displacement of P from O at time t minutes.

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 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 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 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.

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.

The concentration of a drug in the bloodstream of a patient, t hours after the drug has been administered, where t 6, is modelled by the differential equation

where C is measured in micrograms per litre.

Show that the transformation t = e^x transforms equation (I) into the equation

Hence find the general solution for the concentration C at time t hours.

Given that when t = 6, C = 0 and = 36

find the maximum concentration of the drug in the bloodstream of the patient.