Further Differential Equations (DP IB Maths: AI HL)

Exam Questions

5 hours23 questions
1a
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6 marks

Use the Euler method with a step size of 0.1 to find approximations for the values of x and y when t equals 0.5 for each of the following systems of coupled differential equations with the given initial conditions: 

               fraction numerator straight d x over denominator straight d t end fraction equals x squared plus 4 t y 

                  fraction numerator straight d y over denominator straight d t end fraction equals negative 3 x plus y minus t                     

                 x equals 2 comma   y equals 1 space space w h e n space space t equals 0

1b
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6 marks

x with dot on top equals negative x plus straight e to the power of negative t end exponent y 
y with dot on top equals straight e to the power of negative t end exponent x plus y 

x equals 1 comma   y equals negative 1 space space w h e n space space t equals 0

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2a
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6 marks

Consider the following system of differential equations:

                   fraction numerator straight d x over denominator straight d t end fraction equals x plus 2 y 

                  fraction numerator straight d y over denominator straight d t end fraction equals negative 3 x minus 4 y

Find the eigenvalues and corresponding eigenvectors of the matrix  open parentheses table row 1 2 row cell negative 3 end cell cell negative 4 end cell end table close parentheses.

2b
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2 marks

Hence write down the general solution of the system.

2c
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3 marks

When  t equals 0x equals negative 10 and y equals 17.

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

2d
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2 marks

By considering appropriate limits as t rightwards arrow infinity, determine the long-term behaviour of the variables x and y.

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3a
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5 marks

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

                  fraction numerator straight d x over denominator straight d t end fraction equals 4 x plus y

                  fraction numerator straight d y over denominator straight d t end fraction equals negative 5 x minus 2 y

The matrix open parentheses table row 4 1 row cell negative 5 end cell cell negative 2 end cell end table close parentheses has eigenvalues of 3 and negative 1 with corresponding eigenvectors open parentheses table row 1 row cell negative 1 end cell end table close parentheses  and open parentheses table row 1 row cell negative 5 end cell end table close parentheses. Initially x equals 1 and y equals 3.

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

3b
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6 marks

Use the Euler method with a step size of 0.2 to find approximations for the values of x and y when t equals 1.

3c
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5 marks
(i)
Find the percentage error of the approximations from part (b) compared with the exact values of x and y when t equals 1.

(ii)
Comment on the accuracy of the approximations in part (b), and explain how they could be improved.

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4a
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4 marks

For each of the general solutions to a system of coupled differential equations given below,

(i)
sketch the phase portrait for the system

(ii)
state whether the point left parenthesis 0 comma space 0 right parenthesis is a stable equilibrium point or an unstable equilibrium point.

 

               bold italic x equals A e to the power of t open parentheses table row 1 row 1 end table close parentheses plus B e to the power of 2 t end exponent open parentheses table row cell negative 1 end cell row 3 end table close parentheses 
4b
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4 marks

bold italic x equals A e to the power of negative 2 t end exponent open parentheses table row 4 row 1 end table close parentheses plus B e to the power of negative 3 t end exponent open parentheses table row 1 row 5 end table close parentheses

4c
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4 marks

bold italic x equals A e to the power of t open parentheses table row cell negative 1 end cell row 1 end table close parentheses plus B e to the power of negative t end exponent open parentheses table row 4 row 3 end table close parentheses

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5a
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4 marks

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

                      fraction numerator straight d x over denominator straight d t end fraction equals 3 x minus 5 y blank fraction numerator straight d y over denominator straight d t end fraction equals x minus y

where x equals 1and y equals 1 when t equals 0.

The matrix open parentheses table row 3 cell negative 5 end cell row 1 cell negative 1 end cell end table close parentheses has eigenvalues of 1 plus straight i and 1 minus straight i.

(i)
Find the values of ​ fraction numerator straight d x over denominator straight d t end fraction and ​ fraction numerator straight d y over denominator straight d t end fraction at the point left parenthesis 0 comma space 1 right parenthesis.

(ii)
Hence sketch the phase portrait of the system with the given initial condition.
5b
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3 marks

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

fraction numerator straight d x over denominator straight d t end fraction equals negative 3 x minus 5 y      fraction numerator straight d y over denominator straight d t end fraction equals x plus y

with the same initial conditions.

Calculate the eigen values of the matrix open parentheses table row cell negative 3 end cell cell negative 5 end cell row 1 1 end table close parentheses

5c
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2 marks

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

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6a
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2 marks

Scientists have been tracking levels, x and y, 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:

                      fraction numerator straight d x over denominator straight d t end fraction equals x minus 2 y

                      fraction numerator straight d y over denominator straight d t end fraction equals x minus y

Find the values of ​ fraction numerator straight d x over denominator straight d t end fraction and fraction numerator straight d y over denominator straight d t end fraction at the points left parenthesis 1 comma space 0 right parenthesis and left parenthesis 0 comma space 1 right parenthesis.

 

6b
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3 marks

Find the eigenvalues of the matrix open parentheses table row 1 cell negative 2 end cell row 1 cell negative 1 end cell end table close parentheses.

6c
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4 marks

At the start of the study both pollutants are above baseline levels, with x equals 5 and y equals 3.

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

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7a
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6 marks

Two types of bacteria, X and Y, are being grown on a culture plate in a research lab. From past studies of the two bacteria and their interactions, the researchers believe that the growth of the two populations may be represented by the following differential equations

fraction numerator straight d x over denominator straight d t end fraction equals negative 5 x plus 4 y blank fraction numerator straight d y over denominator straight d t end fraction equals negative 8 x plus 7 y

for populations of x thousand and y thousand bacteria of types X and Y respectively. Initially the plate contains 20   000 bacteria of type X and 21   000 of type Y.

Find the eigenvalues and corresponding eigenvectors of the matrix  open parentheses table row cell negative 5 end cell 4 row cell negative 8 end cell 7 end table close parentheses .

7b
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2 marks

Find the values of ​ fraction numerator straight d x over denominator straight d t end fraction and ​ fraction numerator straight d y over denominator straight d t end fraction when t equals 0.

7c
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4 marks

Sketch a possible trajectory for the growth of the two populations of bacteria, being sure to indicate any asymptotic behaviour.

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1a
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6 marks

Consider the following system of coupled differential equations

 x equals negative 3 x plus e to the power of negative 4 t end exponent y

y equals 6 e to the power of negative 2 t end exponent x plus y

with the initial condition x equals 1 comma space y equals 2 when t equals 0.

Use the Euler method with a step size of 0.1 to find approximations for the values of x  and y when t equals 0.5.        

 

1b
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4 marks

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

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2a
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7 marks

Consider the following system of differential equations:

 fraction numerator d x over denominator d y end fraction equals 1 half x minus 2 y

fraction numerator d y over denominator d x end fraction equals x minus 5 over 2 y 

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

2b
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3 marks

When t equals 0 comma space x equals negative 3 and y equals 2.

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

2c
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2 marks

Describe the long-term behaviour of the variables x and y.

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3a
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7 marks

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

fraction numerator d x over denominator d t end fraction equals 3 x minus 2 y

fraction numerator d y over denominator d t end fraction equals 3 x minus 4 y 

The matrix  open parentheses table row 3 cell negative 2 end cell row 3 cell negative 4 end cell end table close parentheses  has eigenvectors open parentheses table row 2 row 1 end table close parentheses  and open parentheses table row 1 row 3 end table close parentheses.  Initially  x equals 7 and y equals 1. 

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

3b
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6 marks

Use the Euler method with a step size of 0.2 to find approximations for the values of x  and y when t equals 1.

3c
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4 marks
(i)
Find the percentage error of the approximations from part (b) compared with the exact values of x and y when t equals 1

(ii)
Explain how the approximations found in part (b) could be improved.

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4a
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4 marks

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

 x equals A e to the power of p t end exponent open parentheses table row 2 row 1 end table close parentheses plus B e to the power of q t end exponent open parentheses table row cell negative 2 end cell row 3 end table close parentheses 

where p and q are real constants. 

For each of the relationships between p and q given below,

i)
sketch the phase portrait for the system 
ii)
state whether the point  is a stable equilibrium point or an unstable equilibrium point.

p less than q less than 0.

 

4b
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4 marks

p less than 0 less than q

4c
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4 marks

0 less than p less than q

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5a
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4 marks

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

fraction numerator d x over denominator d t end fraction equals 3 x minus 5 y           fraction numerator d y over denominator d t end fraction equals x minus y

where x equals 1 and  y equals 1 when t equals 0

The matrix open parentheses table row 3 cell negative 5 end cell row 1 cell negative 1 end cell end table close parentheses has eigenvalues of  1 plus straight i and 1 minus straight i

(i)
Find the values of  fraction numerator d x over denominator d t end fraction and fraction numerator d y over denominator d t end fraction at the point open parentheses 0 comma 1 close parentheses

         

(ii)
Hence sketch the phase portrait of the system with the given initial condition.
5b
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3 marks

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

 fraction numerator d x over denominator d y end fraction equals negative 3 x minus 5 y       fraction numerator d y over denominator d t end fraction equals x plus y 

with the same initial conditions. 

Calculate the eigenvalues of the matrix  open parentheses table row cell negative 3 end cell cell negative 5 end cell row 1 1 end table close parentheses

5c
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2 marks

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

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6a
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2 marks

Scientists have been tracking levels, x and y, 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:

fraction numerator d x over denominator d t end fraction equals x minus 2 y

fraction numerator d y over denominator d t end fraction equals x minus y

Find the values of fraction numerator d x over denominator d t end fraction and fraction numerator d y over denominator d t end fraction at the points  open parentheses 1 comma 0 close parentheses and open parentheses 0 comma 1 close parentheses.

6b
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3 marks

Find the eigenvalues of the matrix  open parentheses table row 1 cell negative 2 end cell row 1 cell negative 1 end cell end table close parentheses

6c
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4 marks

At the start of the study both pollutants are above baseline levels, with x equals 5 and y equals 3.

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

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7a
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4 marks

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  fraction numerator d x over denominator d t end fraction equals 1.9 x minus 0.2 y and fraction numerator d y over denominator d t end fraction equals 0.3 x plus 2.6 y , where x represents the size of the prey population (in thousands) and y 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 open parentheses table row cell 1.9 end cell cell negative 0.2 end cell row cell 0.3 end cell cell 2.6 end cell end table close parentheses are 2.5 and 2, with corresponding eigenvectors open parentheses table row cell negative 1 end cell row 3 end table close parentheses and open parentheses table row cell negative 2 end cell row 1 end table close parentheses, sketch a possible trajectory for the change in the populations of the two animals over time.

7b
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1 mark

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.

7c
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2 marks

It is suggested that the system of equations  fraction numerator d x over denominator d t end fraction equals open parentheses 10 minus 2 y close parentheses x and  fraction numerator d y over denominator d t end fraction equals open parentheses 3 x minus 6 close parentheses y should be used as a model instead, where t is measured in decades (1 decade= 10 years ).

Determine the equilibrium points for the system under this model.

7d
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9 marks
(i)
Use the Euler method with a step size of 0.002 to find approximations for the values of x and y at one-year intervals up to 8 years after the start of the study.

(ii)
Use the values from (d)(i) to sketch a possible trajectory for the change in the populations of the two animals over time, and state what this suggests about the long-term behaviour of the two animal populations under the revised model.

 

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8a
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2 marks

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

 fraction numerator straight d squared x over denominator straight d t squared end fraction plus 7 fraction numerator d x over denominator d t end fraction plus 13 x equals 109 

where fraction numerator d x over denominator d t end fraction and fraction numerator straight d squared x over denominator straight d t squared end fraction represent the particle’s velocity and acceleration respectively.

By letting  y equals fraction numerator d x over denominator d t end fraction,  show that the differential equation above can be written as a system of first order differential equations.

8b
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6 marks

When t equals 0,  the displacement of the particle is zero and the velocity is negative 2 space ms to the power of negative 1 end exponent.

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

i)
displacement
ii)
velocity


of the particle at time t = 0.5 .

8c
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2 marks

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

8d
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1 mark

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

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1a
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6 marks

Consider the following system of coupled differential equations

 x with dot on top equals x squared plus 2 t y

y with dot on top equals negative 6 x plus y plus 14 t 

with the initial condition x equals negative 2 comma space y equals negative 12 when t equals 0. 

Use the Euler method with a step size of 0.1 to find approximations for the values of x  and y when t equals 0.5.        

1b
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5 marks
(i)
Show that the system has a single equilibrium point at time t equals 0 and write down its coordinates.

(ii)
Find the coordinates of the equilibrium points in terms of t. Hence show that, for times t greater than 0, the system has no equilibrium points at which the values of both x and y are non-negative.

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2a
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10 marks

Consider the following system of differential equations:

fraction numerator d x over denominator d t end fraction equals negative 23 over 14 x plus 5 over 7 y

fraction numerator d y over denominator d t end fraction equals 15 over 14 x plus 1 over 7 y 

Given that  x equals 8  and y equals 3 when t equals 0,  

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

2b
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2 marks

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

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3a
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7 marks

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

fraction numerator d x over denominator d t end fraction equals negative 0.3 x minus 2.1 y        fraction numerator d y over denominator d t end fraction equals negative 8.1 x plus 3.3 y 

The matrix open parentheses table row cell negative 0.3 end cell cell negative 2.1 end cell row cell negative 8.1 end cell cell 3.3 end cell end table close parentheses has eigenvalues -3 and 6.  Initially x equals 0  and y equals 3.

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

3b
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6 marks

Use the Euler method with a step size of 0.2 to find approximations for the values of x and y when  t equals 1.

3c
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2 marks

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

3d
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4 marks
(i)
Find the percentage error of the approximations from part (b) compared with the exact values of x and y when t equals 1.

(ii)
Explain how the approximations found in part (b) could be improved.

 

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4a
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5 marks

Given that the matrices open parentheses table row cell negative 0.2 end cell cell 2.4 end cell row cell 2.4 end cell cell 1.2 end cell end table close parentheses comma space open parentheses table row cell 1.36 end cell cell 0.48 end cell row cell 0.48 end cell cell 1.64 end cell end table close parentheses and open parentheses table row cell negative 2.28 end cell cell 0.96 end cell row cell 0.96 end cell cell negative 1.72 end cell end table close parentheses all have open parentheses table row 3 row 4 end table close parentheses and open parentheses table row cell negative 4 end cell row 3 end table close parentheses as eigenvectors,

i)
sketch the phase portrait, and
ii)
state whether the point open parentheses 0 comma 0 close parentheses is a stable equilibrium point or an unstable equilibrium point 

for each of the systems of differential equations given below.

fraction numerator d x over denominator d t end fraction equals negative 0.2 x plus 2.4 y space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 2.4 x plus 1.2 y

4b
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5 marks

fraction numerator d x over denominator d t end fraction equals 1.36 x plus 0.48 y space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 0.48 x plus 1.64 y

4c
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5 marks

fraction numerator d x over denominator d t end fraction equals negative 2.28 x plus 0.96 y space space space space space space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 0.96 x minus 1.72 y

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5a
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7 marks

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

 fraction numerator d x over denominator d t end fraction equals negative x minus 10 y space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 10.1 x minus 3 y 

where x equals 1 and y equals 1 when t equals 0.  

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




Now consider instead the following system of differential equations

fraction numerator d x over denominator d t end fraction equals x minus 10 y space space space space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 10.1 x plus 3 y

with the same initial conditions.

5b
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5 marks

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.

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6a
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4 marks

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

 fraction numerator d x over denominator d y end fraction equals 5 x minus 8.5 y plus 9.5 space space space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals 4 x minus 5 y minus 5 

Find the equations of the lines on which (i) fraction numerator d x over denominator d t end fraction and (ii) fraction numerator d y over denominator d t end fraction are equal to zero, and hence determine the equilibrium point of the system.

6b
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4 marks

Use the substitutions u equals x minus 10 and v equals y minus 7 to rewrite the equations as a system of coupled differential equations in u and v.

6c
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4 marks

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

6d
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4 marks

Initially x equals 1 and y equals 1.

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

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7a
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4 marks

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

 fraction numerator d x over denominator d t end fraction equals open parentheses a minus b y close parentheses x space space space space space space space space space space space space space space space fraction numerator d y over denominator d t end fraction equals open parentheses c x minus d close parentheses y 

where x represents the size of the prey population,   y represents the size of the predator population, and a comma space b comma space c and d 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 x and y will be located.  Give your answers in terms of a comma space b comma space c and d as appropriate.

7b
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6 marks

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

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

7c
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8 marks

Let a equals 13 comma space b equals 5 comma space c equals 4 and  d equals 9,  with x equals 1 and y equals 1 when t equals 0.

By first using the Euler method with a step size of 0.002 to find approximations for the values of x and y between t equals 0 and t equals 0.64,  sketch a phase portrait showing an approximate solution trajectory for the system with the given parameter values and initial conditions.

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8a
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3 marks

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

 x with " on top plus 5 x with dot on top plus 6 x equals 0 

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

8b
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6 marks

When t equals 0 ,  the displacement of the particle is negative 3 m and the velocity is  negative 10 space ms to the power of negative 1 end exponent.

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 t equals 0.5.

8c
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7 marks

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 t equals 0.5 that were found in part (b).

8d
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4 marks

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.

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