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Quiz 12: First order linear differential equations and predator prey models

Question

Question 1

Two populations,

x = x (t)

and

y = y (t)

, satisfy the following differential equations

\begin{array}{l} x^{'} (t) & = 2 x (t) + 3 y (t), \\ y^{'} (t) & = 5 x (t) - 4 y (t) . \end{array}

Eliminate

y

to find a second order linear differential equation in

x

such that

x = x (t)

Your answer is correct.

\begin{array}{l} x^{″} (t) & = 2 x^{'} (t) + 3 y^{'} (t), \\ therefore, x^{″} (t) & = 2 x^{'} (t) + 15 x (t) - 12 y (t), \\ = 2 x^{'} (t) + 15 x (t) - 4 x^{'} (t) + 8 x (t) \\ = - 2 x^{'} (t) + 23 x (t), \\ x^{″} (t) & + 2 x^{'} (t) - 23 x (t) = 0 . \end{array}

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 2

Two populations,

x = x (t)

and

y = y (t)

, satisfy the following differential equations

\begin{array}{l} \frac{d x}{d t} & = 3 x + 2 y, \\ \frac{d y}{d t} & = x + y . \end{array}

Eliminate

y

to find a second order linear differential equation in

x

such that

x = x (t) .

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

\begin{array}{l} \frac{d^{2} x}{d t^{2}} & = 3 \frac{d x}{d t} + 2 \frac{d y}{d t}, \\ \frac{d^{2} x}{d t^{2}} & = 3 \frac{d x}{d t} + 2 x + 2 y, \\ \frac{d^{2} x}{d t^{2}} & = 3 \frac{d x}{d t} + 2 x + \frac{d x}{d t} - 3 x = 0, \\ \frac{d^{2} x}{d t^{2}} & = 4 \frac{d x}{d t} - x, \\ \frac{d^{2} x}{d t^{2}} & - 4 \frac{d x}{d t} + x = 0 . \end{array}

Not correct. Choice (d) is false.

Question 3

Let populations

A

and

B

be represented by

x (t)

and

y (t)

respectively at time

t .

Population

A

’s growth is 4 times itself and it is decreased by 2 times the population of

B .

Population

B

’s growth is 5 times population

A

and it is decreased by 3 times its own population. Which of the systems below model these populations ?

Not correct. Choice (a) is false.

Your answer is correct.

This is a classic predator-prey relationship where population

B

is the predator and population

A

is the prey.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 4

Let populations

A

and

B

be represented by

x (t)

and

y (t)

respectively at time

t .

Population

A

s growth is 2 times itself and it is increased by 3 times the population of

B .

Population

B

s growth is 3 times itself and it is increased by 4 times population

A .

Which of the systems below model these populations ?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

This is a classic symbiotic relationship where each population depends on the other for growth.

Question 5

Find the general solution to the system of linear differential equations

\begin{array}{l} x^{'} (t) & = 4 x (t) - 2 y (t), \\ y^{'} (t) & = 5 x (t) - 3 y (t) . \end{array}

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

First find a differential equation involving

x (t)

only :

\begin{array}{l} x^{″} (t) & = 4 x^{'} (t) - 2 y^{'} (t), \\ x^{″} (t) & = 4 x^{'} (t) - 10 x (t) + 6 y (t), \\ = 4 x^{'} (t) - 10 x (t) + 12 x^{'} (t) - 3 x (t), \\ = x^{'} (t) + 2 x (t), \\ x^{″} (t) & - x^{'} (t) - 2 x (t) = 0 . \end{array}

The auxiliary equation is

k^{2} - k - 2 = (k - 2) (k + 1) = 0

and the general solution is

\begin{array}{l} x (t) & = A e^{2 t} + B e^{- t}, \\ x^{'} (t) & = 2 A e^{2 t} - B e^{- t}, \\ y (t) & = 2 x (t) - \frac{1}{2} x^{'} (t), \\ = 2 A e^{2 t} + 2 B e^{- t} - A e^{2 t} + \frac{1}{2} B e^{- t}, \\ = A e^{2 t} + \frac{5}{2} B e^{- t} . \end{array}

Question 6

Two populations

x = x (t)

and

y = y (t)

satisfy the following differential equations

\begin{array}{l} x^{'} (t) & = 2 x (t) + 3 y (t), \\ y^{'} (t) & = 4 x (t) + 3 y (t) . \end{array}

What is the solution to this system if

x (0) = 3000

and

y (0) = 500 ?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

We first find the differential equation for

x (t)

\begin{array}{l} x^{″} (t) & = 2 x^{'} (t) + 3 y^{'} (t) . \\ therefore, x^{″} (t) & = 2 x^{'} (t) + 12 x (t) + 9 y (t), \\ = 2 x^{'} (t) + 12 x (t) + 3 x^{'} (t) - 6 x (t), \\ = 5 x^{'} (t) + 6 x (t), \\ therefore, x^{″} (t) & - 5 x (t) - 6 x (t) = 0 . \end{array}

It follows that the general solution is

x (t) = A e^{6 t} + B e^{- t}

and

y (t) = \frac{4}{3} A e^{6 t} - B e^{- t} .

Substituting the initial conditions gives

A + B = 3000

and

\frac{4}{3} A - B = 500

which gives

A = B = 1500 .

Question 7

Two populations

x = x (t)

and

y = y (t)

satisfy the following differential equations

\begin{array}{l} \frac{d x}{d t} & = 3 x + 2 y, \\ \frac{d y}{d t} & = 4 x + y . \end{array}

Find the general solution to this system.

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

We first find the differential equation for

x (t)

\begin{array}{l} \frac{d^{2} x}{d t^{2}} & = 3 \frac{d x}{d t} + 2 \frac{d y}{d t}, \\ \frac{d^{2} x}{d t^{2}} & - 3 \frac{d x}{d t} - 8 x - 2 y = 0, \\ \frac{d^{2} x}{d t^{2}} & - 3 \frac{d x}{d t} - 8 x - \frac{d x}{d t} + 3 x = 0, \\ \frac{d^{2} x}{d t^{2}} & - 4 \frac{d x}{d t} - 5 x = 0 . \end{array}

The auxiliary equation is

k^{2} - 4 k - 5 = (k - 5) (k + 1) = 0

hence the general solution is

\begin{array}{l} x (t) & = A e^{5 t} + B e^{- t} . \\ therefore, \frac{d x}{d t} & = 5 A e^{5 t} - B e^{- t}, \\ y (t) & = \frac{1}{2} \frac{d x}{d t} - \frac{3}{2} x \\ = \frac{5}{2} A e^{5 t} - \frac{1}{2} B e^{- t} - \frac{3}{2} A e^{5 t} - \frac{3}{2} B e^{- t} \\ = A e^{5 t} - 2 B e^{- t} . \end{array}

Not correct. Choice (d) is false.

Question 8

Find the particular solution to the system of linear differential equations

\begin{array}{l} x^{'} (t) & = 7 x (t) - 12 y (t), \\ y^{'} (t) & = 4 x (t) - 7 y (t) . \end{array}

Your answer is correct.

We find the differential equation for

x (t)

\begin{array}{l} x^{″} (t) & = 7 x^{'} (t) - 12 y^{'} (t), \\ therefore, x^{″} (t) & = 7 x^{'} (t) - 48 x (t) + 84 y (t), \\ = 7 x^{'} (t) - 48 x (t) - 7 x^{'} (t) + 49 x (t), \\ = x (t), \\ x^{″} (t) & - x (t) = 0 . \end{array}

x (t) = A e^{t} + B e^{- t}

and

y (t) = \frac{1}{2} A e^{t} + \frac{2}{3} B e^{- t} .

Hence

A + B = 1

and

\frac{1}{2} A + \frac{2}{3} B = 1

which gives

A = - 2

and

B = 3 .

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 9

Find the general solution to the system of linear differential equations

\begin{array}{l} x^{'} (t) & = 2 x (t) - 3 y (t), \\ y^{'} (t) & = 3 x (t) + 2 y (t) . \end{array}

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

We find the differential equation for

x (t)

\begin{array}{l} x^{″} (t) & = 2 x^{'} (t) - 3 y^{'} (t), \\ therefore, x^{″} (t) & = 2 x^{'} (t) - 9 x (t) - 6 y (t), \\ = 2 x^{'} (t) - 9 x (t) + 2 x^{'} (t) - 4 x (t), \\ = 4^{'} (t) - 13 x (t), \\ therefore, x^{″} (t) & - 4 x^{'} (t) + 13 x (t) = 0 . \end{array}

So the auxiliary equation

k^{2} - 4 k + 13 = 0

and its roots are

k = 2 \pm 3 \sqrt{- 1}

and the general solution is

\begin{array}{l} x (t) & = e^{2 t} (A sin 3 t + B cos 3 t), \\ x^{'} (t) & = 2 e^{2 t} (A sin 3 t + B cos 3 t) + e^{2 t} (3 A cos 3 t - 3 B sin 3 t) . \\ y (t) & = \frac{2}{3} x (t) - \frac{1}{3} x^{'} (t) \\ = e^{2 t} (B sin 3 t - A cos 3 t) . \end{array}

Question 10

Find the particular solution to the system of linear differential equations

\begin{array}{l} x^{'} (t) & = x (t) - 2 y (t), \\ y^{'} (t) & = 4 x (t) - 3 y (t), \end{array}

where

y (0) = 1

and

y (π) = 2 .

Not correct. Choice (a) is false.

Your answer is correct.

\begin{array}{l} x^{″} (t) & = 2 x^{'} (t) - 3 y^{'} (t) . \\ therefore, x^{″} (t) & = 2 x^{'} (t) - 9 x (t) - 6 y (t), \\ = 2 x^{'} (t) - 9 x (t) + 2 x^{'} (t) - 4 x (t), \\ = 4^{'} (t) - 13 x (t), \\ therefore, x^{″} (t) & = 4 x^{'} (t) + 13 x (t) = 0 . \end{array}

The auxiliary equation is thus

k^{2} - 4 k - 13 = 0

and its roots are

k = 2 \pm 3 \sqrt{- 1}

. The general solution is

\begin{array}{l} x (t) & = e^{2 t} (A sin 3 t + B cos 3 t) . \\ therefore, x^{'} (t) & = 2 e^{2 t} (A sin 3 t - B cos 3 t) + e^{2 t} (3 A cos 3 t - 3 B sin 3 t), \\ y (t) & = \frac{2}{3} x (t) - \frac{1}{3} x^{'} (t) \\ = 2 e^{2 t} (B sin 3 t - A cos 3 t) . \end{array}

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

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Questions:

right first attempt
right
wrong

a)	$x^{″} (t) + 2 x^{'} (t) - 23 x (t) = 0$	b)	$x^{″} (t) - 2 x^{'} (t) + 23 x (t) = 0$
c)	$x^{″} (t) - \frac{2}{3} x^{'} (t) - \frac{7}{3} x (t) = 0$	d)	$x^{″} (t) - 6 x^{'} (t) - x (t) = 0$

a)	$\frac{d^{2} x}{d t^{2}} + 4 \frac{d x}{d t} - x = 0$	b)	$\frac{d^{2} x}{d t^{2}} - 4 \frac{d x}{d t} + 5 x = 0$
c)	$\frac{d^{2} x}{d t^{2}} - 4 \frac{d x}{d t} + x = 0$	d)	$\frac{d^{2} x}{d t^{2}} + 4 \frac{d x}{d t} - 5 x = 0 .$

a)	$x^{'} (t) = 4 x (t) - 2 y (t)$ , $y^{'} (t) = 5 y (t) - 3 x (t)$ .	b)	$x^{'} (t) = 4 x (t) - 2 y (t)$ , $y^{'} (t) = 5 x (t) - 3 y (t)$ .
c)	$x^{'} (t) = 4 x (t) + 2 y (t)$ , $y^{'} (t) = 5 x (t) + 3 y (t)$ .	d)	$x^{'} (t) = 4 x (t) + 2 y (t)$ , $y^{'} (t) = 5 y (t) + 3 x (t)$ .

a)	$x^{'} (t) = 2 x (t) - 3 y (t)$ , $y^{'} (t) = 3 y (t) - 4 x (t)$ .	b)	$x^{'} (t) = 2 x (t) + 3 y (t)$ , $y^{'} (t) = 3 x (t) + 4 y (t)$ .
c)	$x^{'} (t) = 2 x (t) - 3 y (t)$ , $y^{'} (t) = 4 x (t) - 3 y (t)$ .	d)	$x^{'} (t) = 2 x (t) + 3 y (t)$ , $y^{'} (t) = 4 x (t) + 3 y (t)$ .

a)	$x (t) = A e^{2 t} + B e^{t}$ , $y (t) = 3 A e^{2 t} + \frac{3}{2} B e^{t} .$	b)	$x (t) = A e^{- 2 t} + B e^{- t}$ , $y (t) = 5 A e^{2 t} + \frac{5}{2} B e^{t}$ .
c)	$x (t) = A e^{- 2 t} + B e^{t}$ , $y (t) = 6 A e^{- 2 t} + \frac{3}{2} B e^{t} .$	d)	$x (t) = A e^{2 t} + B e^{- t},$ $y (t) = A e^{2 t} + \frac{5}{2} B e^{- t} .$

a)	$x (t) = 3500 e^{2 t} - 500 e^{3 t}$ , $y (t) = 500 e^{3 t} .$	b)	$x (t) = \frac{9500}{3} e^{2 t} - \frac{500}{3} e^{3 t}$ , $y (t) = \frac{500}{3} e^{3 t} .$
c)	$x (t) = 1500 e^{- 6 t} + 1500 e^{t},$ $y (t) = 2000 e^{- 6 t} + 1500 e^{t} .$	d)	$x (t) = 1500 e^{6 t} + 1500 e^{- t},$ $y (t) = 2000 e^{6 t} - 1500 e^{- t} .$

a)	$x (t) = A e^{(2 - \sqrt{15}) t} + B e^{(2 + \sqrt{15}) t}$ $y (t) = \frac{5 - \sqrt{15}}{2} A e^{(2 - \sqrt{15}) t} + \frac{5 + \sqrt{15}}{2} A e^{(2 + \sqrt{15}) t}$	b)	$x (t) = A e^{(- 2 - \sqrt{15}) t} + B e^{(- 2 + \sqrt{15}) t}$ , $y (t) = \frac{1 - \sqrt{15}}{2} A e^{(- 2 - \sqrt{15}) t} + \frac{1 + \sqrt{15}}{2} A e^{(- 2 + \sqrt{15}) t}$ .
c)	$x (t) = A e^{5 t} + B e^{- t}$ , $y (t) = A e^{5 t} + 2 B e^{- t} .$	d)	$x (t) = A e^{- 5 t} + B e^{t},$ $y (t) = 4 A e^{- 5 t} - B e^{t} .$

a)	$x (t) = 3 e^{- t} - 2 e^{t}$ , $y (t) = 2 e^{- t} - e^{t} .$	b)	$x (t) = 2 e^{- t} - e^{t}$ , $y (t) = \frac{2}{3} e^{- t} + \frac{1}{3} e^{t}$ .
c)	$x (t) = \frac{11}{7} sin t + cos t$ , $y (t) = \frac{5}{6} sin t + cos t$ .	d)	$x (t) = \frac{1}{7} e^{7 t} + \frac{6}{7} e^{- 7 t}$ , $y (t) = e^{- 7 t}$ .

a)	$x (t) = A e^{(2 + \sqrt{17}) t} + B e^{(2 - \sqrt{17}) t},$ $y (t) = 3 A e^{2 t} + \frac{3}{2} B e^{t}$ .	b)	$x (t) = A e^{(- 2 + \sqrt{17}) t} + B e^{(- 2 - \sqrt{17}) t},$ $y (t) = - \frac{1}{3} A e^{(- 2 + \sqrt{17}) t} + - \frac{1}{3} B e^{(- 2 - \sqrt{17}) t}$ .
c)	$x (t) = e^{- 2 t} (A sin 3 t + B cos 3 t)$ , $y (t) = e^{- 2 t} ((\frac{4}{3} A + B) sin 3 t - (A - \frac{4}{3} B) cos 3 t)$ .	d)	$x (t) = e^{2 t} (A sin 3 t + B cos 3 t)$ , $y (t) = e^{2 t} (B sin 3 t - A cos 3 t)$ .

a)	$x (t) = e^{- t} (\frac{1}{2} sin 2 t - \frac{5}{2} cos 2 t)$ , $y (t) = e^{2 t} (2 sin 2 t + cos 2 t)$ .	b)	$x (t) = e^{- t} (\frac{1}{2} sin 2 t + \frac{3}{2} cos 2 t)$ , $y (t) = e^{2 t} (2 sin 2 t + cos 2 t)$ .
c)	$x (t) = A e^{(2 + \sqrt{17}) t} + B e^{(2 - \sqrt{17}) t}$ , $y (t) = 3 A e^{2 t} + \frac{3}{2} B e^{t}$ .	d)	$x (t) = A e^{(- 2 + \sqrt{17}) t} + B e^{(- 2 - \sqrt{17}) t}$ , $y (t) = - \frac{1}{3} A e^{(- 2 + \sqrt{17}) t} + - \frac{1}{3} B e^{(- 2 - \sqrt{17}) t}$ .

The University of Sydney - School of Mathematics and Statistics

Quiz 12: First order linear differential equations and predator prey models

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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