Microsoft Word - ECON2016 Unit 6 and 7 Practice Questions.docx Unit6AdditionalPracticeQuestions Find additional practice questions for Unit 6 in preparation for the graded quiz. Solutions will not...

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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 (?
Answered 1 days AfterApr 07, 2022

Solution

Komalavalli answered on Apr 09 2022
14 Votes
SOLUTION.PDF

Answer To This Question Is Available To Download

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