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Question 1

HLPaper 2

A researcher is analyzing the growth of two competing plant species in a controlled environment. The growth rates of species X and species Y can be described by the following system of differential equations:

\frac{d x}{d t}=2 x+y, \quad \frac{d y}{d t}=3 x+4 y.

1.

Find the eigenvalues of the matrix using a calculator.

[5]

Question 2

HLPaper 2

A biologist is studying the interaction between two species in an ecosystem, where the population dynamics can be modeled by the following system of differential equations:

$\frac{d x}{d t}=2 x+y, \quad \frac{d y}{d t}=4 x+3 y .$

1.

Determine the eigenvalues of the system.

[5]

2.

Find the corresponding eigenvectors.

[6]

3.

Classify the origin and describe the behaviour of solutions near it.

[4]

4.

Suggest a practical interpretation of this system in terms of population dynamics.

[3]

Question 3

HLPaper 3

This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.

It is desired to solve the coupled system of differential equations

\dot{x} = 3x + y

\dot{y} = -x + y

The general solution to the coupled system of differential equations is hence given by

\begin{pmatrix} x \\ y \end{pmatrix} = A\begin{pmatrix} 1 \\ -1 \end{pmatrix}e^{2t} + B\begin{pmatrix} t \\ -t + 1 \end{pmatrix}e^{2t}

As $t \to \infty$ the trajectory approaches an asymptote.

1.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

[1]

2.

Show that the matrix

\left( \begin{array}{cc} 3 & 1 \\ -1 & 1 \end{array} \right)

has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

3.

Verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} t \\ -t + 1 \end{array} \right)e^{2t}

is also a solution.

[5]

4.

Find the values of $x$ and $y$ when $t = 2$ .

[2]

5.

Find the equation of this asymptote.

[3]

6.

If initially at $t = 0$ , $x = 20$ , $y = 10$ , find the particular solution.

[3]

7.

Hence, verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 1 \\ -1 \end{array} \right)e^{2t}

is a solution to the above system.

[5]

Question 4

HLPaper 1

The diagram shows the slope field for the differential equation $\frac{dy}{dx} = \sin(x+y), -4 \leq x \leq 5, 0 \leq y \leq 5$ The graphs of the two solutions to the differential equation that pass through points (0, 1) and (0, 3) are shown. For the two solutions given, the local minimum points lie on the straight line L₁.

1.

Find the equation of L₁, giving your answer in the form y = mx + c.

[1]

2.

For the two solutions given, the local maximum points lie on the straight line L₂. Find the equation of L₂.

[1]

Question 5

HLPaper 2

A biologist is studying the population dynamics of two species in a shared habitat. The change in population of species X and species Y over time can be modeled by the following system of differential equations:

$\frac{d x}{d t}=5 x+2 y, \quad \frac{d y}{d t}=x+3 y .$

1.

Calculate the eigenvalues of the system using your calculator.

[5]

2.

Determine if the origin is stable or unstable based on the eigenvalues.

[2]

3.

Describe the shape of the trajectories near the origin based on the eigenvalues.

[5]

Question 6

HLPaper 2

A model describes the dynamics of two chemical concentrations, $x$ and $y$ (in mol/L), in a reactor, governed by the system:

\frac{d x}{d t}=2 x+y, \quad \frac{d y}{d t}=-x+y .

This system has a single eigenvalue.

1.

Show that the matrix $\left(\begin{array}{cc}2 & 1 \\ -1 & 1\end{array}\right)$ has only one eigenvalue, and find it along with an associated eigenvector.

[6]

2.

Verify that $\binom{x}{y}=\binom{1}{-1} e^{1.5 t}$ is a solution to the system.

[4]

3.

Find a second independent solution of the form $\binom{t}{-t+k} e^{1.5 t}$ , and verify it.

[5]

4.

Given initial conditions $x(0)=5, y(0)=0$ , find the particular solution.

[3]

5.

Sketch the phase portrait for $-3 \leq x \leq 3,-3 \leq y \leq 3$ , showing the equilibrium point, an eigenvector direction, and at least two trajectories with arrows.

[4]

Question 7

HLPaper 3

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.

It is desired to solve the coupled system of differential equations

$\dot x = x + 2y - 50$

$\dot y = 2x + y - 40$

where $x$ and $y$ represent the population of two types of symbiotic coral and $t$ is time measured in decades.

1.

Find the equilibrium point for this system.

[2]

2.

If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .

[3]

3.

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

4.

Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .

[3]

5.

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.

[8]

6.

Find the particular solution to the original system, given the initial conditions of part (b).

[2]

7.

Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.

[2]

8.

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

9.

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

[1]

10.

If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.

[2]

11.

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

[2]

Question 8

HLPaper 2

The populations of two symbiotic bacteria, $\mathrm{P}(x)$ and $\mathrm{Q}(y)$ , in a petri dish are modeled by:

\frac{d x}{d t}=2 x+y, \quad \frac{d y}{d t}=x+2 y

where $x$ and $y$ are in millions per $\mathrm{cm}^{2}$ and $t$ is in hours.

1.

Find the general solution using the eigenvalue method.

[8]

2.

Given initial conditions $x(0)=1, y(0)=0.5$ , find the particular solution.

[3]

3.

Determine the asymptotic behavior of the particular solution as $t \rightarrow \infty$ , including the equation of the asymptote.

[3]

4.

Sketch the phase portrait for $0 \leq x \leq 3,0 \leq y \leq 3$ , showing the asymptote and at least two trajectories with arrows.

[4]

Question 9

HLPaper 2

A model for the populations of two symbiotic algae species, $x$ and $y$ (in thousands per $\mathrm{m}^{2}$ ), is given by:

\frac{d x}{d t}=x+2 y-10, \quad \frac{d y}{d t}=2 x+y-15

1.

Find the equilibrium point.

[3]

2.

Use the substitution $X=x-3, Y=y-2$ to transform the system into a homogeneous system, and find its eigenvalues.

[5]

3.

Find the general solution of the original system, given the eigenvalues and eigenvectors from part (b).

[6]

4.

Given initial conditions $x(0)=5, y(0)=3$ , find the particular solution and determine the limit of $\frac{y}{x}$ as $t \rightarrow \infty$ .

[4]

5.

Sketch the phase portrait for $0 \leq x \leq 6,0 \leq y \leq 6$ , showing the equilibrium point and trajectories.

[4]

Question 10

HLPaper 2

An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable x measures the concentration of mercury in micrograms per litre.

1.

Show that the system of coupled first order equations:

$\frac{dx}{dt} = y \frac{dy}{dt} = -2x - 3y$

can be written as the given second order differential equation.

[2]

2.

Find the eigenvalues of the system of coupled first order equations given in part .

[3]

3.

Hence find the exact solution of the second order differential equation.

[1]

4.

Sketch the graph of x against t, labelling the maximum point of the graph with its coordinates.

[2]

5.

Use the model to calculate the total amount of time when fishing should be stopped.

[3]

6.

Write down one reason, with reference to the context, to support this decision.

[1]

AHL 5.17—Phase portrait Questionbank