Further Differential Equations (DP IB Maths: AI HL)

Consider the following system of coupled differential equations

with the initial condition  when .

Use the Euler method with a step size of 0.1 to find approximations for the values of  and  when .        

Show that the system has no equilibrium points other than the origin, for any value of .

Consider the following system of differential equations:

By first finding the eigenvalues and corresponding eigenvectors of an appropriate matrix, determine the general solution of the system.

When  and .

Use the given initial condition to determine the exact solution of the system.

Describe the long-term behaviour of the variables  and .

The rates of change of two variables,  and , are described by the following system of differential equations:

The matrix    has eigenvectors   and .  Initially   and . 

Use the above information to find the exact solution to the system of differential equations.

Use the Euler method with a step size of 0.2 to find approximations for the values of  and when .

Consider a system of coupled differential equations with a general solution given by

where  and  are real constants. 

For each of the relationships between  and given below,

The behaviour of two variables, and , is modelled by the following system of differential equations:

where  and  when . 

The matrix  has eigenvalues of  and . 

Find the values of  and at the point . 

It is suggested that the variables might better be described by the system

with the same initial conditions. 

Calculate the eigenvalues of the matrix 

Hence describe how your phase portrait from part (a)(ii) would change to represent this new system of differential equations.

Scientists have been tracking levels,  and , of two atmospheric pollutants, and recording the levels of each relative to historical baseline figures (so a positive value indicates an amount higher than the baseline and a negative value indicates an amount less than the baseline).  Based on known interactions of the pollutants with each other and with other substances in the atmosphere, the scientists propose modelling the situation with the following system of differential equations:

Find the values of and  at the points   and .

Find the eigenvalues of the matrix  

At the start of the study both pollutants are above baseline levels, with  and .

Use the above information to sketch a phase portrait showing the long-term behaviour of  and .

Scientists are studying populations of a prey species and a predator species within a particular region.  They initially model the two species by the system of differential equations  and  , where represents the size of the prey population (in thousands) and  represents the size of the predator population (in hundreds).  Initially there are 2000 animals in the prey population and 450 in the predator population. 

Given that the eigenvalues of the matrix  are and 2, with corresponding eigenvectors  and , sketch a possible trajectory for the change in the populations of the two animals over time.

Research suggests that neither species will disappear from the region in the foreseeable future. 

Criticise the model above, particularly in light of this research result.

It is suggested that the system of equations   and  should be used as a model instead, where is measured in decades (1 decade= 10 years ).

Determine the equilibrium points for the system under this model.

A particle moves in a straight line, such that its displacement  metres at time  seconds is described by the differential equation

where and  represent the particle’s velocity and acceleration respectively.

By letting  ,  show that the differential equation above can be written as a system of first order differential equations.

When ,  the displacement of the particle is zero and the velocity is .

By applying Euler’s method with a step size of 0.1 to the system of equations found in part (a), along with the given initial condition, find approximations for the

of the particle at time t = 0.5 .

Use the Euler method to determine the long-term stable value of the particle’s displacement.

Use your answer from part (c) to explain why the long-term stable value of the particle’s velocity must be zero.

Consider the following system of coupled differential equations

with the initial condition  when  

Use the Euler method with a step size of 0.1 to find approximations for the values of   and  when         

Consider the following system of differential equations:

Given that    and  when ,  

use a matrix method to determine the exact solution of the system.

Hence determine the long-term ratio of the value of  to the value of .

The rates of change of two variables,  and , are described by the following system of differential equations:

The matrix  has eigenvalues -3 and 6.  Initially   and .

Use the above information to find the exact solution to the system of differential equations.

Use the Euler method with a step size of 0.2 to find approximations for the values of  and  when  .

Compare the ratio of the approximations from part (b) with the ratio of  to  that you would expect in the long term based on your answer to part (a).

Given that the matrices and  all have  and  as eigenvectors,

ii)

state whether the point is a stable equilibrium point or an unstable equilibrium point 

for each of the systems of differential equations given below.

The behaviour of two variables,  and , is modelled by the following system of differential equations:

where  and  when   

Sketch the phase portrait of the system with the given initial condition.

Now consider instead the following system of differential equations

with the same initial conditions.

Describe briefly how the phase portrait for this system would differ from the phase portrait drawn in part (a).  Be sure to justify your answer.

The amounts,  and , of two reactive chemicals in a solution are modelled by the following system of differential equations:

Find the equations of the lines on which (i)  and (ii)  are equal to zero, and hence determine the equilibrium point of the system.

Use the substitutions  and  to rewrite the equations as a system of coupled differential equations in  and .

Determine the nature of the solution trajectories for the system of equations in and   found in part (b).

Initially  and .

Use your answers to parts (a) and (c) to sketch a phase portrait showing the long-term behaviour of  and .  You may take as a given that both  and  remain non-negative for all values of .

Scientists are studying populations of a prey species and a predator species within a particular ecosystem.  They model the two populations by the system of equations

where  represents the size of the prey population,   represents the size of the predator population, and  and  are all positive real parameters.

Write down (i) the coordinates of the equilibrium points of the system, and (ii) the equations of the lines on which the (local) minimum and maximum values of  and will be located.  Give your answers in terms of  and  as appropriate.

Parameter  is sometimes referred to as the ‘prey population growth parameter’, while parameter  is sometimes referred to as the ‘predator population extinction parameter’. 

Using mathematical reasoning, explain briefly (i) why those names are suitable for parameters  and , and (ii) what parameters  and  represent in the model.

Let  and  ,  with  and  when .

By first using the Euler method with a step size of 0.002 to find approximations for the values of  and  between  and ,  sketch a phase portrait showing an approximate solution trajectory for the system with the given parameter values and initial conditions.

A particle moves in a straight line, such that its displacement  metres at time  seconds is described by the differential equation

Show that the second order differential equation above can be rewritten as a system of coupled first order differential equations.

When  ,  the displacement of the particle is  m and the velocity is  .

By applying Euler’s method with a step size of 0.1 to the system of equations found in part (a), find approximations for (i) the displacement and (ii) the velocity of the particle at time .

By first finding the exact solution to the system of equations found in part (a), determine the percentage error of the values for the displacement and velocity at time  that were found in part (b).

Sketch the trajectory of your exact solution from part (c) on a phase diagram, showing the relationship between the particle’s displacement and velocity as time t increases.