Microsoft Word - ECON2016 Unit 6 and 7 Practice Questions.docx
Unit 6 Additional Practice Questions
Find additional practice questions for Unit 6 in preparation for the graded quiz. Solutions
will not be posted but you can email me if you have any specific queries.
1. Find the solution to the initial value problems.
(i) (ii)
(iii)
2. Use the integrating factor method to solve the differential equations.
(i) (ii)
3. Solve the second order differential equations.
(i)
(ii)
(iii)
4. Coal is extracted from a mine at a rate that is proportional to the amount of
coal that remains in the mine. Show that if R is the amount that remains after
t years and is the initial amount of coal in the mine then
, where k is a constant.
Given that after 10 years, 80% of the initial amount of coal remains,
(i) find the value of the constant k, and
(ii) find the time taken for 50% of the initial amount of coal to remain.
2)1(,2 -== y
y
x
dx
dy 1)0(, == - yxe
dx
dy y
1)1(, == yxye
dx
dy x
3)1(,3 3 ==- yx
x
y
dx
dy 0)1(;)1( ==++ - yeyx
dx
dyx x
xey
dx
dy
dx
yd 3
2
2
34 -=++
0 when 0 and 1 ,65 22
2
===-=++ - x
dx
dyyey
dx
dy
dx
yd x
0 when 3 and 1 ,2232
2
====++ - t
dt
dQQeQ
dt
dQ
dt
Qd t
0R
kteRR -= 0
5. The population P(t), at time t, of a prolific
eed of ra
its is such that the
ate of growth of its population is proportional to the square of its
population, so that
(i) Show that
,
where is the population at time .
(ii) “Doomsday is the time when . Argue why doomsday
occurs when .
(iii) Given that and that there are 4 ra
its after 4 months,
estimate the value of the constant k in the above equation. When
does doomsday occur?
6. At time t the population of the world is x where x is treated as a continuous
variable. The time-rate of increase of x due to births is and the time-rate
of decrease of x due to deaths is .
(i) Argue why the differential equation relating x and t is
, where are constants.
(ii) Given that and t is in years, show that the
population of the world will be doubled in approximately 35 years.
7. At time , Osama opened an account and deposited $8,000 in a bank. At
time t days, the interest rate is 2% on the amount P that is present. Osama
withdraws, on a daily basis, $100.
(i) Treating P as a continuous variable, show that
(ii) Solve the differential equation to find P in terms of t.
(iii) Find the time taken for the account to have a balance of $11,000.
(iv) When the account balance was showing $11,000, Osama decides to
withdraw $x on a daily basis. Write down, in terms of x, a new
differential equation that is satisfied by x.
(v) Show that if , then the balance amount will decrease on a
daily basis.
.0 where2 >= kkP
dt
dP
tkP
P
tP
0
0
1
)(
-
=
0P 0=t
¥®)(tP
0
1
kP
t =
20 =P
xa
x
x
dt
dx )( ba -= ba and
04.0 and 06.0 == ba
0=t
.500050 -= P
dt
dP
220>x
8. Solve the following system of differential equations.
subject to .
7043
,3592
21
2
21
1
+--=
+-=
yy
dt
dy
yy
dt
dy
1)0(,0)0( 21 == yy
ECON2016_u XXXXXXXXXXv1
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 1
UNIT 6
Differential Equations
Overview
Any situation where the quantities change over time can be modelled using differential
equations. As a result, differential equations have become indispensable tools in the
modelling and solution of problems in modern society. This unit explores finding
solutions to first order and second order differential equations as well as systems of
differential equations.
Learning Objectives
By the end of this unit, you will be able to:
1. Solve first order differential equations.
2. Solve second order differential equations.
3. Solve systems of differential equations.
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 2
This unit is divided into two sessions as follows:
Session 6.1: First Order Differential Equations
Session 6.2: Second Order Differential Equations
Session 6.3: Systems of Differential Equations
Readings & Resources
Note to Students: Sometimes hyperlinks to resources may not
open when clicked. If any link fails to open, please copy and paste
the link in your
owser to view/download the resource.
Required Readings
Hoy, M., Livernois, J., McKenna, C., Rees, R., & Stengos, T XXXXXXXXXXMathematics
for economics. MIT Press.
Differential Equations
https:
www.youtube.com/watch?v=zid7J4EhZN8
Khan Academy: Differential Equations
https:
www.khanacademy.org/math/ap-calculus-a
ab-diff-
equations/ab-diff-eq-intro/v/differential-equation-introduction
Khan Academy: Method of Undetermined Coefficients
https:
www.khanacademy.org/math/differential-equations/second-order-
differential-equations/undetermined-coefficients/v/undetermined-
coefficients-1
Khan Academy: Second Order Differential Equations
https:
www.khanacademy.org/math/differential-equations/second-order-
differential-equations
Second Order Linear Homogeneous Differential Equations
https:
www.youtube.com/watch?v=UyCwAFQt4v0
Systems of Differential Equations
http:
tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 3
Session 6.1
First Order Differential Equations
Introduction
This session discusses how to solve first order differential equations. It focuses on
different types of first order differential equations such as: ordinary, linear, non-linear
and separable.
First Order Differential Equations: Part 1
Let us start out our discussion on differential equations by reviewing some of what you
may already know. For example, let us find the equation of the curve whose gradient is
given by !"
!#
= 2? and passes through (1,3).
We have !"
!#
= 2? (Differential equation (DE))
? = ∫2? ??
? = ?+ + ? (General solution)
The graph representing this equation is shown below. Note that C is a constant and can
take on any value.
Figure 6.1
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 4
Suppose we have specific values for x and ? . Then we can find values for C. For
example, if ? = 1 and ? = 3 gives then 3 = 1+ + ?
3 = 1 + ?
? = 2
So the equation is ? = ?+ + 2
Most times we want to express how a variable changes over time, t , where time is
considered to be a continuous variable.
e.g. !"!4 + ? = 7
Definitions
Let us formally define some key expressions that we will use in our discussion.
Ordinary Differential Equation contains ordinary derivatives as opposed to partial
derivatives. We can also have partial differential equations e.g. !"
!4
= 3?+?
Differential Equations of Order
i. !"
!#
= 2? 1st Order
ii. !
7"
!47
+ 2 !"
!4
+ ? = 2 2nd Order
iii. !
8"
!48
+ 2 !"
!4
= 4 3rd Order
Linear or Non-Linear Differential Equations
!"
!4
+ ?:? = ? Linear
!"
!4
+ ?+ = ? Non-Linear
Separable Differential Equations
The example shown above is an example of a separable differential equation.
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 5
Example 1
Solve !"!# = 2?
Separate variables ?? = 2? ??
Integrate both sides ∫?? = ∫ 2? ??
? = ?+ + ?
Example 2
Solve !"!4 =
4;:47
"7 , ? = 6 when ? = 0
Separate variables ? ?? = (? + 3?+)??
Integrate both sides ∫ ?+?? = ∫(? + 3?+)??
"
8
:
= 4
7
+
+ ?: + ?
multiply through by 3 ?: = :4
8
+
+ 3?: + 3?
?: = :4
8
+
+ 3?: + ? (General Solution)
Now, ? = 0 ??? ? = 6, this gives 6: = ?
XXXXXXXXXX = ?
So ?: = :4
7
+
+ 3?: XXXXXXXXXXParticular Solution)
? = F:4
7
+
+ 3?: + 216
8
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 6
Example 3
Solve !"!4 = 2?
Separate variables !"
"
= 2 ??
Integrate both sides ∫
!"
"
= ∫2 ??
XXXXXXXXXXln|?| = 2? + ?
? = ?+4;J
? = ?+4. ?J
? = ??+4 where ? = ?M constant
Example 4
Solve !"!4 =
4"
47;N
Separate variables !"4 =
4
47;N
??
Integrate both sides ∫
!"
"
= ∫ 447;N ??
ln|?| = N+ ln(?
XXXXXXXXXXln|?|
ln|?| = ln √? XXXXXXXXXXln|?|
ln|?| = lnP?√?+ + 1P
? = P?√?+ + 1P
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 7
First Order Differential Equations: Part 2
Let us now explore more differential equations and see how we can deal with those that
are not separable.
Integrating Factors
Example 1
Solve !"!4 − 3? = ??
:4 ; ?(0) = 4
Here we cannot separate variables so we need an integrating factor (I.F.)
We can write a general first order linear differential equation as
!"
!4
+ ?(?)? = ?(?)
In our example, ?(?) = −3 and ?(?) = ??:4
To solve this, we multiply by the integrating factor (I.F.) ?∫U(4)!4 . Note that we ignore
the constant of integration, C.
The I.F. turns the left-hand side into the derivative of a product. Now let us solve the
problem.
!"
!4
− 3? = ??:4 , ?(0) = 4 …….(*)
First, find I.F. Here ?(?) = −3
So I.F. = ?∫W: !4
= ?W:4
Next multiply (*) by I.F. to get
?W:4 !"
!4
− 3?W:4? = ??:4. ?W:4
?W:4 !"
!4
− 3?W:4? = ?
!
!4
(?W:4?) = ?
ECON2016: Mathematical Methods of Economics II • UNIT 6 _ 20190225_v1 8
Finally, integrate both sides,
∫ !!4 (?