5.7 Further Differential Equations

1a 1b 2a 2b 2c 3a 3b 3c 4a 4b 4c 5a 5b 5c 6a 6b 6c 7a 7b 7c 7d 8a 8b 8c 8d

Question 1a

Marks: 6

Consider the following system of coupled differential equations

$x = - 3 x + e^{- 4 t} y$

$y = 6 e^{- 2 t} x + y$

with the initial condition $x = 1, y = 2$ when $t = 0$ .

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

x

and

y

when

t = 0.5

Key Concepts

Euler's Method: Coupled Systems

Question 1b

Marks: 4

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

t

Key Concepts

Equilibrium Points
Exponential Functions & Graphs

Question 2a

Marks: 7

Consider the following system of differential equations:

$\frac{d x}{d y} = \frac{1}{2} x - 2 y$

$\frac{d y}{d x} = x - \frac{5}{2} y$

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

Key Concepts

Eigenvalues & Eigenvectors
Solving Coupled Differential Equations

Question 2b

Marks: 3

When $t = 0, x = - 3$ and $y = 2$ .

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

Key Concepts

Solving Coupled Differential Equations

Question 2c

Marks: 2

Describe the long-term behaviour of the variables

x

and

y

Key Concepts

Exponential Functions & Graphs

Question 3a

Marks: 7

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

$\frac{d x}{d t} = 3 x - 2 y$

$\frac{d y}{d t} = 3 x - 4 y$

The matrix $(\begin{matrix} 3 & - 2 \\ 3 & - 4 \end{matrix})$ has eigenvectors $(\begin{matrix} 2 \\ 1 \end{matrix})$ and $(\begin{matrix} 1 \\ 3 \end{matrix})$ . Initially $x = 7$ and $y = 1$ .

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

Key Concepts

Solving Coupled Differential Equations
Eigenvalues & Eigenvectors

Question 3b

Marks: 6

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

x

and

y

when

t = 1

Key Concepts

Euler's Method: Coupled Systems

Question 3c

Marks: 4

Find the percentage error of the approximations from part (b) compared with the exact values of

x

and

y

when

t = 1

ii)

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

Key Concepts

Percentage Error
Euler's Method: Coupled Systems

Question 4a

Marks: 4

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

$x = A e^{p t} (\begin{matrix} 2 \\ 1 \end{matrix}) + B e^{q t} (\begin{matrix} - 2 \\ 3 \end{matrix})$

where $p$ and $q$ are real constants.

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

sketch the phase portrait for the system

ii)

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

(a) $p < q < 0 .$

Key Concepts

Phase Portraits

Question 4b

Marks: 4

b) $p < 0 < q$

Key Concepts

Phase Portraits

Question 4c

Marks: 4

c) $0 < p < q$

Key Concepts

Phase Portraits

Question 5a

Marks: 4

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

$\frac{d x}{d t} = 3 x - 5 y$ $\frac{d y}{d t} = x - y$

where $x = 1$ and $y = 1$ when $t = 0$ .

The matrix $(\begin{matrix} 3 & - 5 \\ 1 & - 1 \end{matrix})$ has eigenvalues of $1 + i$ and $1 - i$ .

(a) (i) Find the values of $\frac{d x}{d t}$ and $\frac{d y}{d t}$ at the point $(0, 1)$ .

(ii) Hence sketch the phase portrait of the system with the given initial condition.

Key Concepts

Phase Portraits

Question 5b

Marks: 3

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

$\frac{d x}{d y} = - 3 x - 5 y$ $\frac{d y}{d t} = x + y$

with the same initial conditions.

b) Calculate the eigenvalues of the matrix $(\begin{matrix} - 3 & - 5 \\ 1 & 1 \end{matrix})$

Key Concepts

Eigenvalues & Eigenvectors

Question 5c

Marks: 2

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

Key Concepts

Phase Portraits

Question 6a

Marks: 2

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:

$\frac{d x}{d t} = x - 2 y$

$\frac{d y}{d t} = x - y$

a) Find the values of $\frac{d x}{d t}$ and $\frac{d y}{d t}$ at the points $(1, 0)$ and $(0, 1)$ .

Question 6b

Marks: 3

Find the eigenvalues of the matrix

(\begin{matrix} 1 & - 2 \\ 1 & - 1 \end{matrix})

Key Concepts

Eigenvalues & Eigenvectors

Question 6c

Marks: 4

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

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

x

and

y

Key Concepts

Phase Portraits

Question 7a

Marks: 4

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 $\frac{d x}{d t} = 1.9 x - 0.2 y$ and $\frac{d y}{d t} = 0.3 x + 2.6 y$ , 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

(\begin{matrix} 1.9 & - 0.2 \\ 0.3 & 2.6 \end{matrix})

are

2.5

and 2, with corresponding eigenvectors

(\begin{matrix} - 1 \\ 3 \end{matrix})

and

(\begin{matrix} - 2 \\ 1 \end{matrix})

, sketch a possible trajectory for the change in the populations of the two animals over time.

Key Concepts

Phase Portraits

Question 7b

Marks: 1

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.

Question 7c

Marks: 2

It is suggested that the system of equations $\frac{d x}{d t} = (10 - 2 y) x$ and $\frac{d y}{d t} = (3 x - 6) y$ 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.

Key Concepts

Equilibrium Points

Question 7d

Marks: 9

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.

Key Concepts

Euler's Method: Second Order
Phase Portraits

Question 8a

Marks: 2

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

$\frac{d^{2} x}{d t^{2}} + 7 \frac{d x}{d t} + 13 x = 109$

where $\frac{d x}{d t}$ and $\frac{d^{2} x}{d t^{2}}$ represent the particle’s velocity and acceleration respectively.

By letting

y = \frac{d x}{d t}

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

Key Concepts

Exact Solutions & Phase Portraits: Second Order

Question 8b

Marks: 6

When $t = 0$ , the displacement of the particle is zero and the velocity is $- 2 {ms}^{- 1}$ .

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

displacement

ii)

velocity

of the particle at time t = 0.5 .

Key Concepts

Euler's Method: Second Order
Displacement, Velocity & Acceleration

Question 8c

Marks: 2

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

Key Concepts

Euler's Method: Second Order

Question 8d

Marks: 1

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

Key Concepts

Displacement, Velocity & Acceleration

DP IB Maths: AI HL

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5.7 Further Differential Equations

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