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University of Sydney
School of Mathematics
and Statistics
© Copyright 2004-2006

Quiz 6: Introduction to Ordinary Differential Equations

Question

Question 1

In a city with a fixed population of $P$ people, the rate of change (with respect to time, $t$ ,) of the number $N$ of people with a certain contagious disease is proportional to the product of the number who have the disease and the number who do not. Which option is a differential equation which describes this situation?

Your answer is correct.

Not correct. Choice (b) is false.

The rate of change of $N$ is proportional to the product of $N$ and $N - P$ , not their sum.

Not correct. Choice (c) is false.

$P$ is a fixed number. It is $N$ that is changing.

Question 2

For which values of $C$ and $n$ (if any) is $y = Cxn$ a solution of the differential equation $xdy-- 3y = 0 dx$ ?

Not correct. Choice (a) is false.

$C = 0$ does give the solution $y = 0$ . However, there are other possibilities.

Your answer is correct.

Not correct. Choice (c) is false.

Try setting $n = 3$ and calculating $dy x --- 3y dx$ .

Not correct. Choice (d) is false.

Using the suggested value $n y = Cx$ , find $-dy- dx$ , substitute it into the given differential equation and factorise.

Question 3

Which function $P (t)$ is a solution to the differential equation $dP- dt = P (1- P)$ ?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

Not correct. Choice (d) is false.

Question 4

The size of a population $P$ is modelled by the differential equation

dP ( P ) -dt = 1.2P 1 - 4200 .

For which values of $P$ is the population increasing?

Not correct. Choice (a) is false.

Remember that $P$ is increasing if its derivative is positive.

Your answer is correct.

Not correct. Choice (c) is false.

These are the values of $P$ for which $dP- dt$ is increasing.

Not correct. Choice (d) is false.

Remember that $P$ is increasing if its derivative is positive.

Question 5

Find the particular solution of the differential equation

dy 2 dx-= xsin (3x ),

given that $y(0) = 0$ .

Not correct. Choice (a) is false.

Note that in this case, $y(0) = - 16$ .

Not correct. Choice (b) is false.

Remember that the derivative of $cosx$ is $- sinx$ .

Not correct. Choice (c) is false.

The question asks for a particular solution, not the general solution.

Your answer is correct.

Question 6

Find the particular solution of the differential equation

dy- 1- dx = x + x+ 1

passing through the point $(1,3)$ . For this particular solution, what is the value of $y$ when $x = 2$ ?

Your answer is correct.

Not correct. Choice (b) is false.

Check that your particular solution is correct!

Not correct. Choice (c) is false.

Check that your particular solution is correct!

Not correct. Choice (d) is false.

Check that your particular solution is correct!

Not correct. Choice (e) is false.

Question 7

Which two functions below satisfy the differential equation

dy 2 dt = 2tantsec t+ sin 2t

There is at least one mistake.
For example, choice (a) should be true.

There is at least one mistake.
For example, choice (b) should be false.

There is at least one mistake.
For example, choice (c) should be false.

There is at least one mistake.
For example, choice (d) should be false.

There is at least one mistake.
For example, choice (e) should be true.

Your answers are correct

True.
False.
False.
False.
True.

Question 8

Find the general solution of the differential equation

d2y 1+-cos3-x- dx2 = cos2x

( $C$ and $D$ are arbitrary constants.)

Not correct. Choice (a) is false.

Antidifferentiation gives $dy ---= tanx + sinx + C dx$ . Now do another antidifferentiation!

Not correct. Choice (b) is false.

Antidifferentiation gives $dy dx-= tanx + sinx + C$ . Now do another antidifferentiation!

Not correct. Choice (c) is false.

Antidifferentiation gives $dy-= tanx+ sinx +C dx$ . Now do another antidifferentiation!

Not correct. Choice (d) is false.

Antidifferentiation gives $dy-= tan x+ sinx+ C dx$ . Now do another antidifferentiation!

Your answer is correct.

Question 9

Find the particular solution of the second order differential equation

d2y = t- e-t dt2

which satisfies the initial conditions $y(0) = 0$ and $′ y (0) = 0$ .

Not correct. Choice (a) is false.

Antidifferentiation gives $dy = 1t2 + e-t + C dt 2$ . Now find the value of $C$ using the condition $y′(0) = 0$ .

Not correct. Choice (b) is false.

Antidifferentiation gives $dy -- = 12t2 + e-t + C dt$ . Now find the value of $C$ using the condition $y′(0) = 0$ .

Your answer is correct.

Not correct. Choice (d) is false.

Antidifferentiation gives $dy 1 2 -t dt = 2t + e + C$ . Now find the value of $C$ using the condition $′ y (0) = 0$ .

Not correct. Choice (e) is false.

Antidifferentiation gives $dy = 1t2 + e-t + C dt 2$ . Now find the value of $C$ using the condition $′ y (0) = 0$ .

Question 10

Given that $x > 0,$ find the particular solution of the differential equation

d2y- 1--3x3- dx2 = x2

satisfying the conditions $′ y(2) = 1$ and $y(1) = 3.$

Your answer is correct.

Not correct. Choice (b) is false.

Antidifferentiation gives $dy-= - x- 1 - 3x2 + C dx 2$ . Now find the value of $C$ using the condition $′ y(2) = 1$ .

Not correct. Choice (c) is false.

Antidifferentiation gives $dy-= - x -1 - 32x2 + C dx$ . Now find the value of $C$ using the condition $y′(2) = 1$ .

Not correct. Choice (d) is false.

Antidifferentiation gives $dy ---= - x-1 - 32x2 + C dx$ . Now find the value of $C$ using the condition $y′(2) = 1$ .