5.7 Further Differential Equations

	Download	View
Medium
Hard
Very Hard

6 marks

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 $t = 0.5$ .

How did you do?

4 marks

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

How did you do?

Did this page help you?

7 marks

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.

How did you do?

3 marks

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

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

How did you do?

2 marks

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

How did you do?

Did this page help you?

7 marks

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.

How did you do?

6 marks

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

How did you do?

4 marks

(i)

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.

How did you do?

Did this page help you?

4 marks

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.

$p < q < 0 .$

How did you do?

4 marks

$p < 0 < q$

How did you do?

4 marks

$0 < p < q$

How did you do?

Did this page help you?

4 marks

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 $1 - i$ .

(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.

How did you do?

3 marks

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.

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

How did you do?

2 marks

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

How did you do?

Did this page help you?

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:

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

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

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

How did you do?

3 marks

Find the eigenvalues of the matrix $(\begin{matrix} 1 & - 2 \\ 1 & - 1 \end{matrix})$

How did you do?

4 marks

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 $y$ .

How did you do?

Did this page help you?

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 $\frac{d x}{d t} = 1.9 x - 0.2 y$ 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.

How did you do?

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.

How did you do?

2 marks

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.

How did you do?

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.

How did you do?

Did this page help you?

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

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

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.

How did you do?

6 marks

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 .

How did you do?

2 marks

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

How did you do?

1 mark

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

How did you do?

Did this page help you?

6 marks

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

\frac{d x}{d t} = x^{2} + 4 t y

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

$x = 2, y = 1 w h e n t = 0$

How did you do?

6 marks

$\dot{x} = - x + e^{- t} y$
$\dot{y} = e^{- t} x + y$

$x = 1, y = - 1 w h e n t = 0$

How did you do?

Did this page help you?

6 marks

Consider the following system of differential equations:

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

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

Find the eigenvalues and corresponding eigenvectors of the matrix $(\begin{matrix} 1 & 2 \\ - 3 & - 4 \end{matrix})$ .

How did you do?

2 marks

Hence write down the general solution of the system.

How did you do?

3 marks

When $t = 0$ , $x = - 10$ and $y = 17$ .

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

How did you do?

2 marks

By considering appropriate limits as $t \to \infty$ , determine the long-term behaviour of the variables $x$ and $y$ .

How did you do?

Did this page help you?

5 marks

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

$\frac{d x}{d t} = 4 x + y$

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

The matrix $(\begin{matrix} 4 & 1 \\ - 5 & - 2 \end{matrix})$ has eigenvalues of $3$ and $- 1$ with corresponding eigenvectors $(\begin{matrix} 1 \\ - 1 \end{matrix})$ and $(\begin{matrix} 1 \\ - 5 \end{matrix})$ . Initially $x = 1$ and $y = 3 .$

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

How did you do?

6 marks

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

How did you do?

5 marks

(i)

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

x

and

y

when

t = 1

(ii)

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

How did you do?

Did this page help you?

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

(0, 0)

is a stable equilibrium point or an unstable equilibrium point.

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

How did you do?

4 marks

$x = A e^{- 2 t} (\begin{matrix} 4 \\ 1 \end{matrix}) + B e^{- 3 t} (\begin{matrix} 1 \\ 5 \end{matrix})$

How did you do?

4 marks

$x = A e^{t} (\begin{matrix} - 1 \\ 1 \end{matrix}) + B e^{- t} (\begin{matrix} 4 \\ 3 \end{matrix})$

How did you do?

Did this page help you?

4 marks

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 $1 - i$ .

(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.

How did you do?

3 marks

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

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

with the same initial conditions.

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

How did you do?

2 marks

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

How did you do?

Did this page help you?

2 marks

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

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

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

How did you do?

3 marks

Find the eigenvalues of the matrix $(\begin{matrix} 1 & - 2 \\ 1 & - 1 \end{matrix})$ .

How did you do?

4 marks

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 $y$ .

How did you do?

Did this page help you?

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

$\frac{d x}{d t} = - 5 x + 4 y \frac{d y}{d t} = - 8 x + 7 y$

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 $(\begin{matrix} - 5 & 4 \\ - 8 & 7 \end{matrix})$ .

How did you do?

2 marks

Find the values of $ \frac{d x}{d t}$ and $ \frac{d y}{d t}$ when $t = 0 .$

How did you do?

4 marks

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

How did you do?

Did this page help you?

6 marks

Consider the following system of coupled differential equations

$\dot{x} = x^{2} + 2 t y$

$\dot{y} = - 6 x + y + 14 t$

with the initial condition $x = - 2, y = - 12$ 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 $t = 0.5 .$

How did you do?

5 marks

(i)

Show that the system has a single equilibrium point at time

t = 0

and write down its coordinates.

(ii)

Find the coordinates of the equilibrium points in terms of

t

. Hence show that, for times

t > 0

, the system has no equilibrium points at which the values of both

x

and

y

are non-negative.

How did you do?

Did this page help you?

10 marks

Consider the following system of differential equations:

$\frac{d x}{d t} = - \frac{23}{14} x + \frac{5}{7} y$

$\frac{d y}{d t} = \frac{15}{14} x + \frac{1}{7} y$

Given that $x = 8$ and $y = 3$ when $t = 0$ ,

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

How did you do?

2 marks

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

How did you do?

Did this page help you?

7 marks

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

$\frac{d x}{d t} = - 0.3 x - 2.1 y$ $\frac{d y}{d t} = - 8.1 x + 3.3 y$

The matrix $(\begin{matrix} - 0.3 & - 2.1 \\ - 8.1 & 3.3 \end{matrix})$ has eigenvalues -3 and 6. Initially $x = 0$ and $y = 3$ .

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

How did you do?

6 marks

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

How did you do?

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).

How did you do?

4 marks

(i)

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.

How did you do?

Did this page help you?

5 marks

Given that the matrices $(\begin{matrix} - 0.2 & 2.4 \\ 2.4 & 1.2 \end{matrix}), (\begin{matrix} 1.36 & 0.48 \\ 0.48 & 1.64 \end{matrix})$ and $(\begin{matrix} - 2.28 & 0.96 \\ 0.96 & - 1.72 \end{matrix})$ all have $(\begin{matrix} 3 \\ 4 \end{matrix})$ and $(\begin{matrix} - 4 \\ 3 \end{matrix})$ as eigenvectors,

sketch the phase portrait, and

ii)

state whether the point

(0, 0)

is a stable equilibrium point or an unstable equilibrium point

for each of the systems of differential equations given below.

\frac{d x}{d t} = - 0.2 x + 2.4 y \frac{d y}{d t} = 2.4 x + 1.2 y

How did you do?

5 marks

$\frac{d x}{d t} = 1.36 x + 0.48 y \frac{d y}{d t} = 0.48 x + 1.64 y$

How did you do?

5 marks

$\frac{d x}{d t} = - 2.28 x + 0.96 y \frac{d y}{d t} = 0.96 x - 1.72 y$

How did you do?

Did this page help you?

7 marks

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

$\frac{d x}{d t} = - x - 10 y \frac{d y}{d t} = 10.1 x - 3 y$

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

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

Now consider instead the following system of differential equations

$\frac{d x}{d t} = x - 10 y \frac{d y}{d t} = 10.1 x + 3 y$

with the same initial conditions.

How did you do?

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.

How did you do?

Did this page help you?

4 marks

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

$\frac{d x}{d y} = 5 x - 8.5 y + 9.5 \frac{d y}{d t} = 4 x - 5 y - 5$

Find the equations of the lines on which (i) $\frac{d x}{d t}$ and (ii) $\frac{d y}{d t}$ are equal to zero, and hence determine the equilibrium point of the system.

How did you do?

4 marks

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

How did you do?

4 marks

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

How did you do?

4 marks

Initially $x = 1$ and $y = 1$ .

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 $t$ .

How did you do?

Did this page help you?

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

$\frac{d x}{d t} = (a - b y) x \frac{d y}{d t} = (c x - d) y$

where $x$ represents the size of the prey population, $y$ represents the size of the predator population, and $a, b, c$ 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 $a, b, c$ and $d$ as appropriate.

How did you do?

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.

How did you do?

8 marks

Let $a = 13, b = 5, c = 4$ and $d = 9$ , with $x = 1$ and $y = 1$ when $t = 0$ .

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

How did you do?

Did this page help you?

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

$\overset{"}{x} + 5 \dot{x} + 6 x = 0$

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

How did you do?

6 marks

When $t = 0$ , the displacement of the particle is $- 3$ m and the velocity is $- 10 {ms}^{- 1}$ .

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 $t = 0.5$ .

How did you do?

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 $t = 0.5$ that were found in part (b).

How did you do?

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.

How did you do?

Did this page help you?

DP IB Maths: AI HL

Topic Questions

5.7 Further Differential Equations