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MAST20029-java代写

时间：2022-11-08

The University of Melbourne
School of Mathematics and Statistics
MAST20029
Engineering Mathematics
Semester 2, 2022
STUDENT NAME:
This compilation has been made in accordance with the provisions of Part VB of the copyright
act for the teaching purposes of the University.
This booklet is for the use of students of the University of Melbourne enrolled in the subject
MAST20029 Engineering Mathematics.
MAST20029 Engineering Mathematics i
MAST20029 Engineering Mathematics
Semester 2 2022
Subject Organisation
MAST20029 Engineering Mathematics is a core mathematics subject that prepares students
for further studies in all branches of Engineering.
This subject is intended only for students pursuing an Engineering Systems major or who
are enrolled in a Master of Engineering degree, who do not wish to take any further study
in Mathematics and Statistics or Physics.
Students who want to supplement their Engineering Systems major with further study
in Mathematics and Statistics or Physics, should seek course advice before enrolling in
MAST20029. In particular, students who want to specialise in Applied Mathematics within
a Mathematics and Statistics Major, should take MAST20009 Vector Calculus, MAST20026
Real Analysis and MAST20030 Differential Equations instead of MAST20029 Engineering
Mathematics.
Syllabus
This subject introduces important mathematical methods required in engineering such as
manipulating vector differential operators, computing multiple integrals and using integral
theorems. A range of ordinary and partial differential equations are solved by a variety of
methods and their solution behaviour is interpreted. The subject also introduces sequences
and series including the concepts of convergence and divergence.
Topics include: Vector calculus, including Gauss’ and Stokes’ Theorems; sequences and
series; Fourier series, Laplace transforms, including convolution; systems of homogeneous
ordinary differential equations, including phase plane and linearisation for nonlinear systems;
second order partial differential equations by separation of variables.
At the completion of this subject, students should be able to
• manipulate vector differential operators
• determine convergence and divergence of sequences and series
• solve ordinary differential equations using Laplace transforms
• sketch phase plane portraits for linear and nonlinear systems of ordinary differential
equations
• represent suitable functions using Fourier series
• solve second order partial differential equations using separation of variables
School of Mathematics and Statistics
ii Course Information 2022
Pre-requisites
One of
• MAST10006 Calculus 2
• MAST10009 Accelerated Mathematics 2
• MAST10019 Calculus Extension Studies
• MAST10021 Calculus 2: Advanced
and one of
• MAST10007 Linear Algebra
• MAST10008 Accelerated Mathematics 1
• MAST10013 UMEP Maths for High Achieving Students
• MAST10018 Linear Algebra Extension Studies
• MAST10022 Linear Algebra: Advanced
Or
• Enrolment in the Master of Engineering
Credit Exclusions
• Students who have completed MAST20009 Vector Calculus or MAST20030 Differential
Equations may not enrol in MAST20029 Engineering Mathematics for credit.
• Concurrent enrolment in MAST20029 Engineering Mathematics and MAST20009 Vec-
tor Calculus is not permitted.
• Concurrent enrolment in MAST20029 Engineering Mathematics and MAST20030 Dif-
ferential Equations is not permitted.
Classes
The subject MAST20029 Engineering Mathematics has
• three one hour lectures per week;
• a one hour practice class per week.
Lectures and practice classes start on the first day of semester. Details of your lecture times
and practice class time are given on your personal timetable in the Student Portal.
The University of Melbourne
MAST20029 Engineering Mathematics iii
Lectures
Students are expected to attend the lectures.
• Tuesday at 11am in JH Mitchell Theatre, Peter Hall Building;
• Wednesday at 3.15pm in JH Mitchell Theatre, Peter Hall Building;
• Thursday at 11am in JH Mitchell Theatre, Peter Hall Building.
Lecturer: Associate Professor Marcus Brazil (Subject Coordinator), Room 5.11, Electrical
and Electronic Engineering.
The lectures will include live polls. To participate in these you will need to have a mobile
phone or some other internet-enabled device. You will be able to take part in the polls at
the URL: pollev/marcusbrazil098
Subject Resources
Textbook
The recommended textbook for MAST20029 is:
• E Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley, 2011.
The textbook is recommended for extra reading and problems but is not compulsory. Any
earlier editions of the textbook are also suitable. It can be purchased as an e-book from
Wiley Direct at www.wileydirect.com.au . The textbook can be borrowed from the Eastern
Resource Centre (ERC) library and the University Library also has an electronic copy of the
book.
The Kreyszig textbook covers all topics in MAST20029 except for sequences and series.
There are many first year calculus textbooks in the ERC library that can be used as a
reference for the sequence and series section of Engineering Mathematics.
Lecture Notes
All students are required to have a copy of the MAST20029 Engineering Mathematics Lecture
Notes, which can be downloaded from the MAST20029 website.
These notes contain the theory, diagrams, and statement of the questions to be covered in
lectures. Students are expected to bring these partial lecture notes to all lectures, and fill in
the working of examples in the gaps provided. This can be done on a tablet/ipad or on a
printed version of the notes.
Practice Class Sheets
A practice class question sheet to be worked on during the practice class will be issued before
each practice class. Students are expected to attempt the questions during the class. Full
solutions to the questions will be provided sometime after the practice class.
The practice class in the first week of semester covers revision material from first year math-
ematics subjects that is essential pre-requisite knowledge for MAST20029. From week 2
onwards, the practice class will be based on the previous week of lectures.
You should aim to complete all the questions on the practice class sheets during semester.
School of Mathematics and Statistics
iv Course Information 2022
Problem Sheets
All students are required to have a copy of the MAST20029 Engineering Mathematics Prob-
lem Sheet Booklet, which can be downloaded from the MAST20029 website.
This problem booklet is for you to work on during your private study time to learn and
practice key concepts, prepare for your practice classes, and to revise for the mid-semester
test and final exam. There are six problem sheets with answers in this booklet corresponding
to the six major topics covered in lectures. At the end of each week you will be advised on
the LMS which questions should be attempted before attending your next practice class.
Whilst working through the problem sheets, make a list of any concepts or topics you are
having difficulty with and ask for help during the individual consultation sessions. Tutors
will not be discussing the specific questions in the problem booklet in the practice classes.
You should aim to complete all the questions in the problem booklet during semester.
Website
All material to do with the assignments, mid-semester test, exam, practice classes, consul-
tation roster and other announcements will be available from the subject LMS (Learning
Management System), at the address:
canvas.lms.unimelb.edu.au
Expectations
In MAST20029 Engineering Mathematics you are expected to:
• Attend all lectures (either in person or watching the video of the lecture), and take
notes and participate in class activities during lectures.
• Attend all practice classes (either in person or online via Zoom), participate in group
work in practice classes, and complete all practice class exercises.
• Work through the problem booklet outside of class in your own time. You should try
to keep up-to-date with the problem booklet questions, and attempt all questions from
the problem booklet before the exam.
• Read the weekly modules, which will be posted on the LMS every Friday afternoon
during the semester.
• Check the announcements on the LMS at least twice per week to make sure you do
not miss any important subject information.
• Complete all assignments on time.
• Seek help when you need it during consultation sessions.
In total, you are expected to dedicate around 170 hours to this subject, including classes.
This equates to an average of about 9 hours of additional study, outside of class, per week
over 14 weeks.
The University of Melbourne
MAST20029 Engineering Mathematics v
MATLAB
Students are expected to use the software package MATLAB throughout the subject Engi-
neering Mathematics to complete questions on problem sheets and assignments.
Detailed information about MATLAB is provided on pages xiv-xxiv of this booklet.
Assessment
The assessment is composed of three major components:
• A three-hour exam worth 70% at the end of the semester.
• A mid semester test worth 15% on about Tuesday 6th September (exact date and time to
be confirmed closer to the date).
• Three assignments worth 5% each, due as follows:
(1) 4.00pm on Monday 22nd August;
(2) 4.00pm on Monday 19th September;
(3) 4.00pm on Monday 17th October.
The mid semester test and exam will be conducted online, via Zoom. Full details will be
posted on the LMS, closer to the dates.
Hurdle requirement: Students must pass the assessment during semester to pass the sub-
ject. That is, students must obtain a mark of at least 15% out of 30% for the combined mid
semester test and assignments to pass the subject.
Assignments
• The assignments will be posted on the LMS one week before the due date.
• Your assignment must be handwritten, but this includes digitally handwritten documents
using an ipad or a tablet and stylus, which have then been saved as a pdf. Typed or typeset
assignments will not be accepted (unless you have received permission beforehand due to a
medical reason). The only exceptions to this are the answers to the MATLAB questions,
which should be screenshots of your MATLAB session, as explained on the cover sheet.
• Your assignment must be scanned or photographed and compiled into a single pdf document
to be uploaded to the LMS website.
• Extensions
Students with medical certificates or other appropriate supporting documentation can apply
to Marcus Brazil for an extension of up to 3 days after the assignment deadline by emailing
her a scanned copy of the documentation. The assignment should be submitted via the LMS.
Do not use the Student Portal to apply for special consideration for the assignments; this
online application is for the final Engineering Maths exam only.
• Late assignments
The following applies to an assignment submitted late, where no extension has been sought
and granted.
Between one day late and three days late: A mark penalty of 20% of the assignment total
will be deducted from the student’s result for each day the assignment is submitted late. For
example, if the assignment deadline is 4.00pm on Monday, then an assignment submitted
after 4.00pm on Monday and before 4.00pm on Tuesday is one day late, so 20% of the
assignment total will be deducted.
School of Mathematics and Statistics
vi Course Information 2022
More than three days late: Assignment is not accepted and a mark of zero is awarded for the
assignment.
Mid-Semester Test
Special consideration
Students with medical certificates or other appropriate supporting documentation can apply
to Marcus Brazil for special consideration for the mid-semester test. You must request special
consideration no later than 3 days after the date of the mid-semester test. You then have a
further 5 days in which to provide a medical certificate or other requested documentation.
Applications lodged after these time limits will not be accepted.
Do not use the Student Portal to apply for special consideration for the mid-semester test;
this online application is for the final Engineering Maths exam only.
Special consideration outcomes
The following applies to students who have applied for special consideration or who have
missed the mid-semester test for any reason.
• Students who sat the test but were sick during the test will be allowed to sit the alterna-
tive sitting of the mid-semester test. Any student who sits the original test as well as the
alternative test will be given the mark from the alternative test, even if it is lower than their
mark in the original test.
• Students unable to sit the test due to illness or any other acceptable reason will be allowed
to sit the alternative sitting of the mid-semester test.
• Students who did not sit the test and did not give any reasonable excuse will receive a mark
of zero for the mid-semester test.
Special Consideration for Exam and Whole Subject
If something major goes wrong during semester or you are sick during the examination
period, you should consider applying for Special Consideration through the Student Portal.
You must submit your online special consideration application no later than 4 days after
the date of the final exam in MAST20029 Engineering Mathematics. You will also need to
submit the completed Health Professional Report (HPR) Form with your online application.
The HPR Form can only be completed by the professional using the form provided.
For more details see the Special Consideration menu item on the website:
http://ask.unimelb.edu.au/app/home
The University of Melbourne
MAST20029 Engineering Mathematics vii
Calculators, Formula Sheets and Dictionaries
Students are not permitted to use calculators, dictionaries, computers, or any electronic
resources in the end of semester exam.
The formula sheet on pages x to xiv of this booklet will be provided in the end of semester
exam, as part of the exam paper.
Assessment in this subject concentrates on the testing of concepts and the ability to conduct
procedures in simple cases. There is no formal requirement to possess a calculator for this
subject. Nonetheless, there are some questions on the problem sheets for which calculator
usage is appropriate. If you have a calculator or an equivalent app on your phone or laptop,
then you will find it useful occasionally.
Getting Help
Lecturers and tutors have consultation hours when they will help you on an individual basis
with questions from the MAST20029 Engineering Mathematics lecture notes, problem sheets
and practice class sheets. Attendance is on a voluntary basis. Details will be provided on
the MAST20029 LMS web site.
School of Mathematics and Statistics
viii Course Information 2022
Lecture Outline
This lecture outline is a guide only. The material to be covered in each lecture may vary
slightly from the following table.
Vector Calculus
1. Vector fields, div and curl operators.
2. Double integrals over general regions, change of order of integration.
3. Double integrals - change of variables, polar coordinates.
4. Triple integrals - change of variables, cylindrical coordinates.
5. Triple integrals in cylindrical coordinates and spherical coordinates.
6. Parametrisation of paths. Line integrals.
7. Work integrals. Conservative fields.
8. Integrals of scalar functions over surfaces, surface area, mass.
9. Integrals of vector functions over surfaces, flux.
10. Gauss’ divergence theorem.
11. Stokes’ theorem.
Systems of First Order Ordinary Differential Equations
12. Systems of linear homogenous ODEs. Solve using eigenvalues and eigenvectors.
13. 2× 2 systems examples, phase space, critical points, phase portraits.
14. Phase portraits of linear systems.
15. Phase portraits of linear systems. Non-linear coupled first order ODEs.
16. Non-linear coupled first order ODEs, linearisation, phase portraits.
Laplace Transforms
17. Laplace transforms. Table of transforms.
18. Inversion of transforms using tables. Solution of ODEs (single and systems).
19. Mid-Semester test (no lecture on that day)
20. S-shifting theorem. Step functions.
21. T-shifting theorem. Impulse and Dirac delta functions.
22. Convolution theorems. Solution of integral equations.
Series
23. Infinite series - partial sums, geometric series, harmonic series.
24. Integral test. Comparison test. Ratio test.
25. Leibniz test. Power series - radius and interval of convergence.
26. Power series (continued). Taylor polynomials.
27. Taylor series, errors.
The University of Melbourne
MAST20029 Engineering Mathematics ix
Fourier Series
28. Periodic functions, Fourier series, Euler’s formulae, energy density,
Parseval’s identity.
29. Fourier series for odd and even functions, periodic extensions.
30. Application of Fourier series to ODEs.
31. Fourier integrals, odd/even functions, applications to ODEs.
Second Order Partial Differential Equations
32. Examples and classification of second order PDEs.
33. Separation of variables for Laplace’s equation.
34. Separation of variables for the wave equation.
35. Separation of variables for the diffusion equation.
36. Revision lecture
School of Mathematics and Statistics
x Course Information 2022
MAST20029 Engineering Mathematics Formulae Sheet
1. Change of Variable of Integration in 2D∫∫
R
f(x, y) dxdy =
∫∫
R∗
f(x(u, v), y(u, v))|J(u, v)| dudv
2. Transformation to Polar Coordinates
x = r cos θ, y = r sin θ, J(r, θ) = r
3. Change of Variable of Integration in 3D∫∫∫
V
f(x, y, z) dxdydz =
∫∫∫
V ∗
F (u, v, w)|J(u, v, w)| dudvdw
4. Transformation to Cylindrical Coordinates
x = r cos θ, y = r sin θ, z = z, J(r, θ, z) = r
5. Transformation to Spherical Coordinates
x = r cos θ sinφ, y = r sin θ sinφ, z = r cosφ, J(r, θ, φ) = r2 sinφ
6. Line Integrals∫
C
f(x, y, z) ds =
∫ b
a
f(x(t), y(t), z(t))
√
x′(t)2 + y′(t)2 + z′(t)2 dt
7. Work Integrals ∫
C
F(x, y, z) · dr =
∫ b
a
F1
dx
dt
+ F2
dy
dt
+ F3
dz
dt
dt
8. Surface Integrals∫∫
S
g(x, y, z) dS =
∫∫
R
g(x, y, f(x, y))
√
f 2x + f
2
y + 1 dxdy
9. Flux Integrals For a surface with upward unit normal,∫∫
S
F · nˆ dS =
∫∫
R
−F1fx − F2fy + F3 dydx
The University of Melbourne
MAST20029 Engineering Mathematics xi
10. Gauss’ (Divergence) Theorem∫∫∫
V
∇ · F dV =
∫∫
S
F · nˆ dS
11. Stokes’ Theorem ∫∫
S
(∇× F) · nˆ dS =
∫
C
F · dr
12. Laplace Transforms
1. f(t) F (s) =
∫ ∞
0
f(t)e−st dt (Definition of Transform)
2. 1
1
s
3. tn
n!
sn+1
4. eat
1
s− a
5. sin(at)
a
s2 + a2
6. cos(at)
s
s2 + a2
7. sinh(at)
a
s2 − a2
8. cosh(at)
s
s2 − a2
9. δ(t− a) e−as (a ≥ 0)
10. f ′(t) sF (s)− f(0)
11. f ′′(t) s2F (s)− sf(0)− f ′(0)
12. f (n)(t) snF (s)−
n−1∑
k=0
sn−1−kf (k)(0)
13.
∫ t
0
f(τ) dτ
F (s)
s
14. e−atf(t) F (s+ a) (s-Shifting Theorem)
15. f(t− a)u(t− a) e−asF (s) (a > 0, t-Shifting Theorem)
16.
∫ t
0
f(τ) g(t− τ) dτ F (s)G(s) (Convolution)
School of Mathematics and Statistics
xii Course Information 2022
13. Standard Limits
(i) lim
n→∞
1
np
= 0 (p > 0) (ii) lim
n→∞
rn = 0 (|r| < 1)
(iii) lim
n→∞
a1/n = 1 (a > 0) (iv) lim
n→∞
n1/n = 1
(v) lim
n→∞
an
n!
= 0 (all a) (vi) lim
n→∞
logen
np
= 0 (p > 0)
(vii) lim
n→∞
(
1 +
a
n
)n
= ea (all a) (viii) lim
n→∞
np
an
= 0 (all p, a > 1)
14. The Generalised Harmonic Series (p-Series)
∞∑
n=1
1
np
is
{
convergent if p > 1
divergent if p ≤ 1
15. Geometric Series
∞∑
n=0
arn is
{
convergent if |r| < 1
divergent if |r| ≥ 1
16. Taylor Polynomial
Pn(x) = f(a) + f
′(a)(x− a) + f
′′(a)
2!
(x− a)2 + · · ·+ f
(n)(a)
n!
(x− a)n
17. The Remainder in Taylor’s Theorem
Rn(x) =
f (n+1)(c)
(n+ 1)!
(x− a)(n+1) where c lies between a and x
18. Fourier Series Formulae
f(t) = a0 +
∞∑
n=1
[an cos(nωt) + bn sin(nωt)] , ω =
2pi
T
=
pi
L
a0 =
1
2L
∫ L
−L
f(t) dt
an =
1
L
∫ L
−L
f(t) cos(nωt) dt
bn =
1
L
∫ L
−L
f(t) sin(nωt) dt
19. Parseval’s Identity for Energy Density
1
T
∫ T/2
−T/2
f 2(t) dt = a20 +
1
2
∞∑
n=1
(a2n + b
2
n)
The University of Melbourne
MAST20029 Engineering Mathematics xiii
20. Fourier Cosine Series for Even Functions
f(t) = a0 +
∞∑
n=1
an cos(nωt)
a0 =
1
L
∫ L
0
f(t) dt
an =
2
L
∫ L
0
f(t) cos(nωt) dt
21. Fourier Sine Series for Odd Functions
f(t) =
∞∑
n=1
bn sin(nωt)
bn =
2
L
∫ L
0
f(t) sin(nωt) dt
22. Fourier Integral Formulae
f(t) =
∫ ∞
0
A(ω) cos(ωt) +B(ω) sin(ωt) dω
A(ω) =
1
pi
∫ ∞
−∞
f(t) cos(ωt)dt
B(ω) =
1
pi
∫ ∞
−∞
f(t) sin(ωt)dt
23. Fourier Cosine Integrals for Even Functions
f(t) =
∫ ∞
0
A(ω) cos(ωt) dω
A(ω) =
2
pi
∫ ∞
0
f(t) cos(ωt) dt
24. Fourier Sine Integrals for Odd Functions
f(t) =
∫ ∞
0
B(ω) sin(ωt) dω
B(ω) =
2
pi
∫ ∞
0
f(t) sin(ωt) dt
25. Complex Exponential Formulae
coshx = 1
2
(ex + e−x) sinhx = 1
2
(ex − e−x)
eix = cosx+ i sinx
cos z = 1
2
(eiz + e−iz) sin z = 1
2i
(eiz − e−iz)
School of Mathematics and Statistics
xiv Course Information 2022
26. Standard Integrals∫
sinx dx = − cosx+ C
∫
cosx dx = sinx+ C∫
secx dx = loge | secx+ tanx|+ C
∫
cosecx dx = loge |cosecx− cotx|+ C∫
sec2 x dx = tanx+ C
∫
cosec 2x dx = − cotx+ C∫
sinhx dx = coshx+ C
∫
coshx dx = sinhx+ C∫
sech 2x dx = tanhx+ C
∫
cosech 2x dx = − cothx+ C∫
1√
a2 − x2 dx = arcsin
(x
a
)
+ C
∫
1√
x2 + a2
dx = arcsinh
(x
a
)
+ C∫ −1√
a2 − x2 dx = arccos
(x
a
)
+ C
∫
1√
x2 − a2 dx = arccosh
(x
a
)
+ C∫
1
a2 + x2
dx =
1
a
arctan
(x
a
)
+ C
∫
1
a2 − x2 dx =
1
a
arctanh
(x
a
)
+ C
where a > 0 is constant and C is an arbitrary constant of integration.
27. Trigonometric and Hyperbolic Formulae
cos2 x+ sin2 x = 1 cosh2 x− sinh2 x = 1
1 + tan2 x = sec2 x 1− tanh2 x = sech 2x
cot2 x+ 1 = cosec 2x coth2 x− 1 = cosech 2x
cos(2x) = cos2 x− sin2 x cosh(2x) = cosh2 x+ sinh2 x
cos(2x) = 2 cos2 x− 1 cosh(2x) = 2 cosh2 x− 1
cos(2x) = 1− 2 sin2 x cosh(2x) = 1 + 2 sinh2 x
sin(2x) = 2 sinx cosx sinh(2x) = 2 sinhx coshx
sin(x+ y) = sin x cos y + cosx sin y sinh(x+ y) = sinhx cosh y + coshx sinh y
cos(x+ y) = cos x cos y − sinx sin y cosh(x+ y) = cosh x cosh y + sinhx sinh y
sinx sin y = 1
2
[cos(x− y)− cos(x+ y)] sinhx sinh y = 1
2
[cosh(x+ y)− cosh(x− y)]
cosx cos y = 1
2
[cos(x− y) + cos(x+ y)] coshx cosh y = 1
2
[cosh(x+ y) + cosh(x− y)]
sinx cos y = 1
2
[sin(x− y) + sin(x+ y)] sinh x cosh y = 1
2
[sinh(x+ y) + sinh(x− y)]
The University of Melbourne
MAST20029 Engineering Mathematics xv
MATLAB Assessment Requirements and Commands
1. Using MATLAB
• MATLAB is installed on computers available for student use throughout the Univer-
sity. Engineering Mathematics students can use the Wilson Lab, Thompson Lab and
Nanson Lab in the Peter Hall building, whenever the labs are vacant.
• In Engineering Maths, we use MATLAB to perform numerical calculations and sym-
bolic manipulations. The Symbolic Math Toolbox is used for symbolic manipulations.
MATLAB (with the symbolic toolbox) is installed on all University Computers and
comes with the Student Edition of MATLAB.
• MATLAB can also be accessed using
(a) “myUniApps” which lets you use University-licensed software on your own com-
puter or tablet via a server in the cloud. Go to http : //myuniapps.unimelb.edu.au/.
(b) MATLAB Online which runs in your browser. See MAST20029 website for details.
• Enter commands in the MATLAB Command Window. You are in the MATLAB
command window when you see the MATLAB command prompt >>.
• To put your name and student number or comments in the command window use
a percentage sign before the comment. For example: type % name, student number,
comment (see sample over page).
• To print from within the MATLAB command window, highlight what you want to
print and choose Print Selection under the File menu.
• To print a figure produced by MATLAB choose Print under the File menu on the
Figure or use print screen commands.
• Using a semi colon at the end of a MATLAB command stops the output of that
command being displayed on the screen.
• Use the up and down arrows in the command window to scroll through previously
typed commands, so you can reuse them or edit them.
• Plots and graphs appear in a separate figure window.
• Use the help menu and search under functions: mathematics to get a list of all
the mathematical function names. For example: cosh(2*x) is the hyperbolic cosine
function cosh(2x) and heaviside(t-pi) is the step function u(t− pi).
2. MATLAB Assessment Requirements
The requirements for MATLAB in Engineering Maths assessment are:
• You must include your name and student number in a comment in your code. MAT-
LAB code submitted without these details will be given zero marks.
• You must include a printout of all MATLAB code and outputs printed from within
MATLAB or as a screen-shot showing your work and the MATLAB command window
heading.
• You may write MATLAB scripts and then run them if you prefer, as long as the
complete script is also included in your printed output.
• Only submit working code. Use the Clear Command Window under the Edit menu
to remove all incorrect working and copy the correct commands from the Command
History Window.
A sample of what is required for assessment is given below:
School of Mathematics and Statistics
xvi Course Information 2022
The University of Melbourne
MAST20029 Engineering Mathematics xvii
3. Taylortool Assessment Requirements
The requirements for using Taylortool in Engineering Maths assessment are:
• You must include your name and student number in a comment in your code when
calling Taylortool. MATLAB code submitted without these details will be given zero
marks.
• The function f , degree of polynomial N , expansion point a, x range as well as graphs
of the function and Taylor polynomial.
A sample of what is required for assessment is given below:
The following commands were compiled using version R2021a of MATLAB. These
commands also work in earlier versions of MATLAB.
4. Vectors and Matrices
To create a row vector a =
[
1 5 2 3
]
type:
>> a=[1 5 2 3]
The vector a will be echoed on the screen.
To create the vector a but not display it on the screen we use a semicolon, namely:
>> a=[1 5 2 3];
To see your vector, type
>> a
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xviii Course Information 2022
To enter a column vector, semi colons are required between the rows. To create the
column vector
c =

2
−5
2
6

we use:
>> c=[2; -5; 2; 6]
We can combine the row/column vector commands to define matrices. To define the
matrix
A =

1 2 3
4 5 6
7 8 9
10 11 12

we type:
>> A=[1 2 3; 4 5 6; 7 8 9; 10 11 12]
To find the inverse, rank, determinant and eigenvalues of the matrix
M =
 4 0 1−2 1 0
−2 0 1

use the following commands:
>> M=[4 0 1; -2 1 0; -2 0 1]
>> inv(M)
>> rank(M)
>> det(M)
>> eig(M)
To find the eigenvalues and eigenvectors of the matrix M, we can use the command:
>> [v,d]=eig(M)
The columns of the matrix v gives the eigenvectors normalised to length 1. The diagonal
entries of the matrix d give the eigenvalues.
• It is important to be aware of the different types of matrix multiplication available.
>> A*A %used to multiply scalars or multiply square matrices
>> A.*A %used to square each entry in the vector or matrix
For example, we can use:
>> M=[4 0 1; -2 1 0; -2 0 1]
>> MM=M*M
>> MMM=M.*M
The University of Melbourne
MAST20029 Engineering Mathematics xix
which produces
>> MM =
14 0 5
-10 1 -2
-10 0 -1
>> MMM =
16 0 1
4 1 0
4 0 1
5. Plotting Graphs
To enter the following data[
x
y
]
=
[
0 1 2 3 4 5 6 7 8 9 10
0.5 1.5 2.5 3.5 4.5 5.5 4.5 3.5 2.5 1.5 0.5
]
into two vectors x and y, we can type:
>> x=[0 1 2 3 4 5 6 7 8 9 10]
>> y=[0.5 1.5 2.5 3.5 4.5 5.5 4.5 3.5 2.5 1.5 0.5]
As x consists of the numbers 0 to 10 in steps of 1, we can also enter it as
>> x=[0:1:10]
To plot y versus x, we use the command
>> plot(x,y)
This command plots a line through the data points with x on the horizontal axis and
y on the vertical axis. The plot appears in a separate figure window.
To plot the data with red stars for each point (no line), use
>> plot(x,y,’r*’)
For a list of plotting colours and line types, type help plot. I To plot more than one
line on the same graph, use the hold (on/off) function as follows.
>> x=[0:1:10];
>> y=[0.5 1.5 2.5 3.5 4.5 5.5 4.5 3.5 2.5 1.5 0.5];
>> z=[3 2 7 3.5 1 5.2 6 8 .9 .3 5];
>> p1=plot(x,y,’r-’)
>> hold on
>> p2=plot(x,z,’k-.’)
>> hold off
We can also add a title and axis labels to the graph using the following commands:
School of Mathematics and Statistics
xx Course Information 2022
>> title(’Plot of Speed vs. Distance’)
>> xlabel(’Distance’)
>> ylabel(’Speed’)
To plot curves y = f(x), we can parametrise the curve using the variable t. For
example, to plot y = x2 from (−1, 1) to (2, 4), we can use
>>t=[-1:.01:2];
>>plot(t,t.^2)
An easy way to sketch a parametrised curve in 2D is to use the ezplot command. To
plot y = x2 from (−1, 1) to (2, 4), we can use
>>ezplot(’t’,’t^2’,[-1,2])
This command automatically labels the x and y axes and puts a heading on the graph
of x = t and y = t2.
To plot a curve in 3D we can use the plot3 command. Plot the helix with parametri-
sation
r(t) = cos ti + sin tj + 2tk
from (1, 0, 0) to (1, 0, 4pi).
>>t= 0:pi/50:2*pi;
>>plot3(cos(t),sin(t),2*t)
6. Derivatives and Integrals using the Symbolic Toolbox
To do calculations on symbolic expressions, we use the MATLAB symbolic toolbox.
To see a list of the commands available type help symbolic. To use commands from the
toolbox you need to first declare the names of your variables, for example to declare
the variable x we type:
>> syms x
To differentiate and integrate the function f(x) =
x
1 + x2
we use the diff and int
commands.
>> syms x
>> f=x/(1+x^2)
>> diff(f)
>> int(f)
We can evaluate definite integrals by adding terminals to the int command:
>> syms x
>> f=x/(1+x^2)
>> int(f,’x’,0,1)
The University of Melbourne
MAST20029 Engineering Mathematics xxi
We can also evaluate double and triple integrals using the int command multiple times.
To evaluate ∫ 1
0
∫ x2
0
xy2 dy dx
we use
>> syms x y
>> int(int(x*y^2,’y’,0,x^2),’x’,0,1)
To evaluate ∫ 1
0
∫ z2
0
∫ y2
0
xyz dx dy dz
we use
>> syms x y z
>> int(int(int(x*y*z,’x’,0,y^2),’y’,0,z^2),’z’,0,1)
7. Differential Equations
We can use the Symbolic Toolbox and the dsolve command to solve single differential
equations and systems of differential equations. We can find the general solution or
the solution subject to initial conditions.
To find the general solution of the second order differential equation
y′′ + y′ = 2t
we use:
>> syms y(t)
>> eqn = diff(y,t,2) + diff(y,t) == 2*t;
>> S = dsolve(eqn)
To find the solution of the differential equation subject to the initial conditions y(0) =
1, y′(0) = 0, we use:
>> syms y(t)
>> eqn = diff(y,t,2) + diff(y,t) == 2*t;
>> Dy = diff(y,t);
>> cond = [y(0)==1, Dy(0)==0];
>> ySol(t) = dsolve(eqn,cond)
To find the general solution of the system of differential equations
dx
dt
= x+ 3y
dy
dt
= 2x+ 2y
we use:
>> syms x(t) y(t)
>> eqns = [diff(x,t) == x + 3*y, diff(y,t) == 2*x + 2*y];
>> S = dsolve(eqns)
School of Mathematics and Statistics
xxii Course Information 2022
You then access the solutions by addressing the elements of the structure:
>> xSol(t) = S.x
>> ySol(t) = S.y
To find the solution of the same system subject to the initial conditions x(0) =
0, y(0) = 1, we use:
>> syms x(t) y(t)
>> eqns = [diff(x,t) == x + 3*y, diff(y,t) == 2*x + 2*y];
>> cond = [x(0)==0, y(0)==1];
>> S = dsolve(eqns,cond)
8. Other Calculations using the Symbolic Toolbox
We can also use the Symbolic Toolbox to factorise polynomials over the real numbers:
>> syms x
>> f=x^2+2*x+1
>> factor(f)
or expand them:
>> syms x
>> g=(x+1)^2
>> expand(g)
To simplify an expression we use the simplify command:
>> syms x
>> simplify(sin(x)^2 + cos(x)^2)
which gives 1.
Sometimes the MATLAB output is hard to read even after using the simplify command.
The pretty command can be useful to reformat symbolic expressions and write them
as fractions.
>> syms x
>> pretty(1/x+1/x^2)
To compute the Laplace transform of f(t) = sin(3t) we use
>> syms t
>> laplace(sin(3*t))
To compute the inverse Laplace transform of F (s) =
1
(s− 2)(s− 1) we use
>> syms s
>> F=1/(s-2)/(s-1)
>> ilaplace(F)
The University of Melbourne
MAST20029 Engineering Mathematics xxiii
To compute the limit lim
n→∞
loge(n
2)
n
we use the limit command:
>> syms n
>> a=log(n^2)/n
>> limit(a,n,inf)
To compute the Taylor polynomial of a function we use the taylor command. The
word ’ExpansionPoint’ can be omitted for a function of one variable. To compute the
4th degree polynomial (5 terms in the polynomial including the constant term) about
a = −1 for f(x) = xex we can use either of the commands below.
>> syms x f
>> f=x*exp(x)
>> taylor(f,x,’ExpansionPoint’,-1,’Order’,5)
>> taylor(f,x,-1,’Order’,5)
The diff command gives f (5)(x), which is needed for the remainder function
R4(x) =
f (5)(c)(x− x0)
5!
.
>> syms x f
>> f=x*exp(x)
>> diff(f,5)
9. Taylortool
Taylor polynomials can be sketched using the MATLAB applet taylortool. To start
the interactive package type:
>> taylortool
You will need to enter the function f , number of terms N , expansion point a and the
x-interval required in the display window.
10. Phase Portraits
Phase portraits can be plotted by running a MATLAB m-file. The latest m-file for
versions MATLAB 2018b to MATLAB 2021a is available on the MAST20029 website.
For earlier MATLAB versions, download the MATLAB m-file from the website
http://math.rice.edu/%7Edfield/index.html
11. Fourier Series
Consider the saw tooth wave defined by
f(t) =
{
0, −pi < t < 0
t, 0 < t < pi
with f(t) = f(t+ 2pi).
School of Mathematics and Statistics
xxiv Course Information 2022
We can use MATLAB to see how the Fourier series
f(t) = a0 +
N∑
n=1
(an cos(nωt) + bn sin(nωt))
approximates the function.
Below is code to plot the original function as well as the Fourier series approximation
for the case N = 5 in the same picture. For a better approximation, replace 5 in the
“for” loop by a larger integer. Comments have been added to some lines to explain
the code.
>> t=-pi:0.01:0;
>> x=zeros(size(t)); %define zero vector with same size as t
>> plot(t,x)
>> axis([-4 4 -1 4])
>> hold on
>> t=0:0.01:pi;
>> y=t;
>> plot(t,y)
>> t=-pi:0.01:pi;
>> f=pi/4; %a_0 term
>> for n=1:5 %compute sin and cos terms
sinterm = sin(n*t)*(-1)^(n+1)/n; %nth sin term
if n/2 == round(n/2) %check if n is even
costerm = 0; %nth cos term (n even)
else %case where n is odd
costerm = -(2/pi)*cos(n*t)/n^2;
end
f = f + sinterm + costerm; %add terms to f
end
>> plot(t,f)
Alternatively, you can sum series
b∑
n=a
f using the symbolic toolbox commands
>> syms n
>> symsum(f,n,a,b)
where f defines the terms in the series with respect to the symbolic variable n, and a
and b are the terminals for the sum.
The University of Melbourne
MAST20029 Engineering Mathematics
Sheet 1: Vector Calculus
1. Consider the velocity field:
v(x, y) = −2yi + 2xj
(a) Sketch the velocity field at the points (1, 0), (2, 0), (0, 1), (0, 2), (−1, 0), (−2, 0),
(0,−1) and (0,−2).
(b) Find ∇ · v and ∇× v.
2. Consider the velocity field:
v(x, y) = x2yi− 2y2j
(a) Sketch the velocity field at the points (0,−1), (−1, 1), (−2,−1), (1, 2), (0, 1),
(2, 1), and (1, 1).
(b) Find ∇ · v and ∇× v.
3. Find the divergence of the following vector fields:
(a) V(x, y, z) = yzi + xzj + xyk
(b) V(x, y, z) = xi + (y + cosx)j + (z + exy)k
4. Find the curl of the following vector fields:
(a) V(x, y, z) = yzi + xzj + xyk
(b) V(x, y, z) = (x2 + y2 + z2)(3i + 4j + 5k)
5. Evaluate the following double integrals:
(a)
∫ 3
0
∫ x
0
4− y2 dydx (b)
∫ 10
1
∫ 1/y
0
yexy dxdy
6. Using double integrals, find the area of the
(a) region enclosed by x = 1− y2 and y = −x− 1
(b) ellipse
x2
a2
+
y2
b2
= 1 if a > 0, b > 0
1
7. For each of the following double integrals:
(a)
∫ 1
0
∫ √y
y
dxdy (b)
∫ 2
0
∫ √4−y2
−
√
4−y2
y dxdy
(c)
∫ 2
0
∫ 4
y2
yex
2
dxdy (d)
∫ 1
0
∫ ex
1
dydx
(i) Sketch the region of integration in the xy-plane.
(ii) Obtain an equivalent double integral with the order of integration reversed.
(iii) Evaluate the reversed integral.
(iv) Show how you would use MATLAB to evaluate the integral.
8. Let R be the region in the xy-plane bounded by the circle x2 + y2 = 4.
(a) Sketch the region R and describe it in terms of polar coordinates.
(b) By changing the variables to polar coordinates, evaluate∫∫
R
10− 2x2 − 2y2 dydx
9. Let R be the region bounded by the circles x2 + y2 = 1 and x2 + y2 = 4.
(a) Sketch the region R and describe it in terms of polar coordinates.
(b) By changing the variables to polar coordinates, evaluate∫∫
R
3x+ 8y2 dydx
10. Evaluate the following double integrals by changing the variables to polar coordinates.
(a)
∫ 2
−2
∫ √4−y2
−
√
4−y2
sin(x2 + y2) dxdy (b)
∫ 2
0
∫ √4−x2
0
ex
2+y2 dydx
11. Let the solid region V be the tetrahedron bounded by the planes
x = 0, y = 0, z = 0 and 3 = 3x+ 3y + z
(a) Sketch V and describe it in cartesian coordinates.
(b) Find the volume of V .
2
12. Let the solid region V be the tetrahedron bounded by the planes
x = 0, y = 0, z = 0 and 6 = 6x+ 3y + z
(a) Sketch V and describe it in cartesian coordinates.
(b) Find the volume of V .
13. Let V be the solid region bounded above by the paraboloid z = 4− x2− y2 and below
by the plane z = 0.
(a) Sketch the region V and describe it in cylindrical coordinates.
(b) Use cylindrical coordinates to find∫∫∫
V
y dzdydx
(c) Use MATLAB to evaluate the triple integral in part (b).
14. Let V be the solid region such that x2 + y2 ≤ 4, x ≥ 0, and bounded by the planes
z = 0 and z = 4.
(a) Sketch the region V and describe it in cylindrical coordinates.
(b) If the density (mass per unit volume) of V is given by
ρ(x, y, z) = xz
use cylindrical coordinates to find the mass of V .
15. Let B be the region within the cylinder x2 + y2 = 1 that is above the xy plane and
below the cone z =
√
x2 + y2.
(a) Sketch the region B.
(b) Using cylindrical coordinates, evaluate∫∫∫
B
z dxdydz
16. Let V be the solid region inside the sphere x2+y2+z2 = a2 that lies above the xy-plane.
(a) Sketch the region V and describe it in spherical coordinates.
(b) Find the volume of V .
3
17. Let V be the solid region such that x2 + y2 + z2 ≤ 4 and y ≥ 0.
(a) Sketch the region V and describe it in spherical coordinates.
(b) If the density (mass per unit volume) of V is given by
ρ(x, y, z) = x2 + y2
use spherical coordinates to find the mass of V .
18. Let V be the solid region bounded above by the sphere r = 1 and below by the cone
φ = pi/3.
(a) Sketch the region V and describe it in spherical coordinates.
(b) Find the volume of V .
(c) Check your answer to part (b) using MATLAB.
19. Give parametrisations for the following curves in terms of a parameter t, with t in-
creasing. Sketch the curve, using arrows to show the direction for increasing t.
(a) The straight line from (−1, 2) to (3, 3).
(b) The portion of the circle x2 + y2 = 4 traversed anticlockwise from (2, 0) to (0, 2).
(c) The portion of the circle x2 + y2 = 4 traversed clockwise from (−2, 0) to (0, 2).
(d) The part of the ellipse
x2
4
+
y2
9
= 1 that lies above the line y = 0, traversed
clockwise.
(e) The part of the parabola y = 2x− x2 from (0, 0) to (2, 0).
(f) The part of the parabola y = 2x− x2 from (2, 0) to (0, 0).
20. Let C be the part of the circle x2 +y2 = 9 that lies between the points (0, 3) and (3, 0),
oriented clockwise.
(a) Evaluate ∫
C
x2y ds
(b) Show how you would use MATLAB to sketch this curve.
21. Let C be the curve y = x3/2 from the point (1, 1) to (4, 8).
(a) Find a parametrisation for C.
(b) Find the length of C.
(c) Show how you would use MATLAB to sketch this curve.
4
22. Consider the helix with parametrisation
r(t) = 2 cos ti + 2 sin tj + tk
Let C be the coil from (−2, 0, pi) to (2, 0, 2pi).
(a) Sketch this part of the helix, using arrows to show the direction for increasing t.
(b) Suppose the density (mass per unit length) of the coil is
ρ(x, y, z) = y2 + 2z
Find the mass of the coil.
23. Let C be the curve with parametrisation
x(t) = t3, y(t) = −t, z(t) = t2, 1 ≤ t ≤ 2
and let the force field be
F(x, y, z) = xi− yzj + z2k
Find the work done by F in moving a particle along C.
24. Consider the force field
F(x, y) = (xy + 2y2)i + (3x2 + y)j
Find the work done by F to move a particle from (0,0) to (1,1) along
(a) y = x2
(b) the y-axis to (0, 1) and then along the line y = 1
(c) path (b) and then returns to the origin along path (a).
25. Consider the following vector fields F. Which vector fields are conservative? For each
conservative vector field, find a scalar function φ such that F = ∇φ.
(a) F(x, y, z) = 2xi + 3yj + 4zk
(b) F(x, y) = y cosxi + x sin yj
(c) F(x, y, z) = 2xyezi + x2ezj + (x2yez + z2)k
26. In each case, show that F is a conservative vector field and find a scalar function φ
such that F = ∇φ. Evaluate
∫
C
F · dr along paths joining (1,−2, 1) to (3, 1, 4).
(a) F(x, y, z) = (2xyz + sinx)i + x2zj + x2yk
(b) F(x, y, z) = (2xy + z3)i + x2j + 3xz2k
5
27. Evaluate
∫
c
F · dr along the given path.
(a) F(x, y, z) = ex sin yi + ex cos yj + z2k
c = (
√
t, t3, e
√
t), 0 ≤ t ≤ 1.
(b) F(x, y, z) = (xy2 + 3x2y)i + (x3 + x2y)j
c is the curve consisting of line segments from (1,1) to (0,2) to (3,0).
28. Let the surface S be the part of the paraboloid
z = 16− x2 − y2
that lies above the xy-plane. Sketch S and find its surface area.
29. Let the surface S be the part of the cone
z =
√
x2 + y2
that lies between the xy-plane and the plane z = 4. Sketch S and find its surface area.
30. Let the surface S be the part of the plane 2x− y + z = 3 that lies above the triangle
in the xy-plane that is bounded by the lines y = 0, x = 1 and y = x. Find the total
mass of S if its density (mass per unit area) is given by
ρ(x, y, z) = xy + z
31. Let the surface S be the part of the paraboloid z = x2+y2 that lies between the planes
z = 4 and z = 9. Sketch S and then evaluate∫∫
S
xy
z
dS
32. Let S be the part of the cone z2 = x2 + y2 that lies between the planes z = 1 and
z = 2. Let S be oriented with outward unit normal. Find the flux of the vector field
F(x, y, z) = −z2k
across S.
33. Let S be the surface defined by the unit sphere x2 + y2 + z2 = 1, and let S be oriented
with outward unit normal. Find the flux of the vector field
F(x, y, z) = zk
across S.
6
34. Let S be the part of the paraboloid z = 1− x2 − y2 that lies above the xy-plane, and
let S be oriented with outward unit normal. Find the flux of the vector field
F(x, y, z) = xi + yj + zk
across S.
35. Use Gauss’ theorem to find the flux of the vector field
F(x, y, z) = x2yi + 2xzj + yz3k
across the surface S of the cuboid V defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 3.
Let S be oriented with outward unit normal.
36. Use Gauss’ theorem to find the flux of the vector field
F(x, y, z) = yi + xj + zk
across the surface S of the region V formed by the portion of the cylinder x2 + y2 ≤ 4
that lies between the plane z = 0 and the paraboloid z = x2 + y2. Let S be oriented
with outward unit normal.
37. Use Gauss’ theorem to find the flux of the vector field
F(x, y, z) = xi + yj + z2k
across the surface S of the sphere with radius a, centred at the origin, and oriented
with outward unit normal.
38. Use Gauss’ theorem to find the flux of the field
F(x, y, z) = x3zi + y3zj + xyk
across the surface S of the sphere with equation x2+y2+z2 = 9, oriented with outward
unit normal.
39. Use Gauss’ theorem to find the flux of the vector field
F(x, y, z) = yi + zj + zk
across the surface S of the tetrahedron V with sides formed by the planes x = 0, y = 0,
z = 0 and x+ 2y + 3z = 6. Let S be oriented with outward unit normal.
40. Use Stokes’ theorem to evaluate ∫
C
F · dr
where the vector field is
F(x, y, z) = yi− xj + yxk
and C is the boundary of the disk S defined by the equations x2 + y2 = 1 and z = 1.
Assume S is oriented with upward unit normal.
7
41. Verify Stokes’ theorem when the vector field is
F(x, y, z) = (x− y)2i + 2zj + x2k
and S is the cone z =
√
x2 + y2 with the circle x2 + y2 = 4, z = 2 as its boundary. Let
S be oriented with outward unit normal.
42. Verify Stokes’ theorem when the vector field is
F(x, y, z) = yi− xj + yxk
and S is the paraboloid z = x2 + y2 with the circle x2 + y2 = 1, z = 1 as its boundary.
Let S be oriented with outward unit normal.
43. Let S be the surface given by
z = 3−
√
x2 + y2, 1 ≤ z ≤ 3.
Assume S is oriented using the outward unit normal.
(a) Sketch the surface S.
(b) Let F(x, y, z) = −yz3i + 3xj + x5k. Evaluate the surface integral∫∫
S
(∇× F) · nˆ dS
using Stokes’ theorem and
(i) an appropriate line integral;
(ii) the simplest surface for S.
ADDITIONAL EXERCISES
Kreyszig 10th Edition
Problem Set 9.4 (page 380): Q. 15 - 20
Problem Set 9.8 (page 405): Q. 1 - 6, 8
Problem Set 9.9 (page 408): Q. 2, 4 - 8
Problem Set 10.1 (page 418): Q. 2 - 11
Problem Set 10.3 (page 432): Q. 2 - 8, 9 - 11
Problem Set 10.6 (page 450): Q. 1 - 10, 12 - 16
Problem Set 10.7 (page 457): Q. 1 - 8, 9 - 18
Problem Set 10.9 (page 468): Q. 1 - 10, 13 - 20
8
MAST20029 Engineering Mathematics
Sheet 2: Systems of First Order ODE’s and Phase Plane
1. (a) Find the eigenvalues and eigenvectors of the matrix[
4 −3
5 −4
]
Use MATLAB to verify your answers.
(b) Using part (a), find the general solution of the system
dx
dt
= 4x− 3y, dy
dt
= 5x− 4y
(c) Verify your answer to part (b) by showing that it satisfies the original system of
differential equations.
2. (a) Using eigenvalues and eigenvectors, find the general solution to
dx
dt
= x− y, dy
dt
= x+ y
(b) Verify your answer to part (a) by showing that it satisfies the original system of
differential equations.
3. Using eigenvalues and eigenvectors, find the general solution of the following coupled
differential equations.
(a)
x˙ = x+ 9y
y˙ = −x− 5y (b)
x˙ = x+ y
y˙ = −x+ 3y
4. Using eigenvalues and eigenvectors, solve the following systems of first order differential
equations subject to the given initial conditions.
(a)
x˙ = x+ y
y˙ = 4x− 2y
x(0) = 1
y(0) = 2
(b)
x˙ = x+ 2y
y˙ = −1
2
x+ y
x(0) = 2
y(0) = 3
In each case, use MATLAB to check your solution.
9
5. Consider the following linear systems and their general solutions:
(a)
dx
dt
= 6x− 2y, dy
dt
= 4x+ 2y
with general solution(
x(t)
y(t)
)
= α1
(
cos(2t)
cos(2t) + sin(2t)
)
e4t + α2
(
sin(2t)
sin(2t)− cos(2t)
)
e4t
(b)
dx
dt
= 5x+ 2y,
dy
dt
= 4x+ 3y
with general solution(
x(t)
y(t)
)
= α1
(
1
1
)
e7t + α2
(
1
−2
)
et
(c)
dx
dt
= 3x− y, dy
dt
= 6x− 4y
with general solution(
x(t)
y(t)
)
= α1
(
1
6
)
e−3t + α2
(
1
1
)
e2t
(d)
dx
dt
= −2x− y, dy
dt
= x
with general solution(
x(t)
y(t)
)
= α1
( −1
1
)
e−t + α2
( −t+ 1
t
)
e−t
For each of the linear systems
(i) Sketch the phase portrait near the critical point at the origin.
(ii) Discuss the type and stability of the critical point.
6. Consider the following linear systems:
(a)
dx
dt
= −y, dy
dt
= x
(b)
dx
dt
= −10x− y, dy
dt
= 15x− 2y
For each of the linear systems
(i) Obtain the general solution.
(ii) Sketch the phase portrait near the critical point at the origin.
(iii) Discuss the type and stability of the critical point at the origin.
(iv) Find the solution satisfying x(0) = 2 and y(0) = 7. Use MATLAB to sketch the
corresponding orbit.
10
7. Consider the following linear systems:
(a)
dx
dt
= −6x− 2y, dy
dt
= 4x− 2y
(b)
dx
dt
= −2x, dy
dt
= −2y
For each of the linear systems
(i) Obtain the general solution.
(ii) Sketch the phase portrait near the critical point at the origin.
(iii) Discuss the type and stability of the critical point at the origin.
(iv) Find the solution satisfying x(0) = 0 and y(0) = 2.
8. Consider the nonlinear system
dx
dt
= y,
dy
dt
= x2 − 4x+ y
(a) Find the critical points.
(b) Determine whether or not the linear system can be used to approximate the non-
linear system near each critical point.
(c) If the linear system can be used to approximate the non-linear system, determine
the type and stability of the critical point.
(d) Use MATLAB to sketch the global phase portrait for the nonlinear system in the
region −4 ≤ x ≤ 8 and −4 ≤ y ≤ 4.
9. Consider the nonlinear system
dx
dt
= y,
dy
dt
= − sinx− y
(a) Find the critical points.
(b) Determine whether or not the linear system can be used to approximate the non-
linear system near each critical point.
(c) If the linear system can be used to approximate the non-linear system, determine
the type and stability of the critical point.
(d) Use MATLAB to sketch the global phase portrait for the nonlinear system in the
region −10 ≤ x ≤ 10 and −4 ≤ y ≤ 4.
11
10. Consider the nonlinear system
dx
dt
= y,
dy
dt
= −x+ (1− x2)y
(a) Find the critical points.
(b) Determine whether or not the linear system can be used to approximate the non-
linear system near each critical point.
(c) If the linear system can be used to approximate the non-linear system, determine
the type and stability of the critical point.
(d) Use MATLAB to sketch the global phase portrait for the nonlinear system in the
region −4 ≤ x ≤ 4 and −4 ≤ y ≤ 4.
11. Consider the nonlinear system
dx
dt
= y,
dy
dt
= −x3 + 2x
(a) Find the critical points.
(b) Determine whether or not the linear system can be used to approximate the non-
linear system near each critical point.
(c) If the linear system can be used to approximate the non-linear system, determine
the type and stability of the critical point.
(d) Use MATLAB to sketch the global phase portrait for the nonlinear system in the
region −4 ≤ x ≤ 4 and −4 ≤ y ≤ 4.
ADDITIONAL EXERCISES
Kreyszig 10th Edition
Problem Set 4.3 (page 147): Q. 1 - 6, 8, 10 - 15
Problem Set 4.4 (page 151): Q. 1 - 10
Problem Set 4.5 (page 159): Q. 4 - 8
12
MAST20029 Engineering Mathematics
Sheet 3: Laplace Transforms
1. Using the integral definition, find the Laplace transforms of
(a) f(t) =
{
kt/c, 0 ≤ t ≤ c
0, t > c
(b) f(t) =
{
k, 0 < a ≤ t ≤ b
0, elsewhere
(c) f(t) = sinh(at) (d) f(t) = cos(at)
2. Use the tables to find the Laplace transforms of
(a) f(t) = 2t2 + 6− 3e4t (b) f(t) = A sin(ωt) +B cosh(kt)
(c) f(t) = t4 + t5 (d) f(t) = cos2 t
In each case use MATLAB to verify your answers.
3. Use the tables to find the inverse Laplace transforms of
(a) F (s) =
3
s2
(b) F (s) =
4
s2 + 16
(c) F (s) =
s
s2 + 9
(d) F (s) =
5s
s2 − 25
In each case use MATLAB to verify your answers.
4. Use partial fractions to find the inverse Laplace transforms of
(a) F (s) =
s
s2 + 3s− 4 (b) F (s) =
1
(s− 2)(s+ 1)
5. Using the s-shifting theorem, find the Laplace transforms of
(a) f(t) = 3te2t (b) f(t) = 5e−6t sin(4t)
(c) f(t) = 2e−t sinh(3t) (d) f(t) = 2 sinh t cos t
6. Use the s-shifting theorem to find the inverse Laplace transforms of
(a) F (s) =
3
(s+ 2)2
(b) F (s) =
1
(s− 2)3
(c) F (s) =
1
(s− 4)2 + 1 (d) F (s) =
s+ 2
(s+ 2)2 + 4
7. Using the t-shifting theorem, find the Laplace transforms of
(a) f(t) = cosh(t− 1)u(t− 1) (b) f(t) = tu(t− pi)
Use MATLAB to verify your answer to part (b).
13
8. Consider the following functions:
(a) f(t) =
{
et if 0 ≤ t < 2
0 elsewhere
(b) f(t) =
{
cos(pit) if 1 ≤ t < 4
0 elsewhere
(c) f(t) =

1 if 0 ≤ t < 5
t if 5 ≤ t < 10
0 if t ≥ 10
(d) f(t) =

0 if 0 ≤ t < 1
sin(pit) if 1 ≤ t < 2
0 if 2 ≤ t < 3
e−2t if t ≥ 3
Represent f using step functions. Hence find the Laplace transform of f .
9. Use the t-shifting theorem to find the inverse Laplace transforms of
(a) F (s) =
e−3s
s
(b) F (s) =
(
5s+ 4
s2
)
e−2s
10. Find the inverse Laplace transforms of
(a) F (s) =
e−3s
s+ 8
(b) F (s) =
e−2s
s2 − 3s− 4
(c) F (s) =
s+ 6
(s+ 2)2 + 4
(d) F (s) =
2
s(s2 + 4)
(e) F (s) = 2 +
e−s
s2 − 16 (f) F (s) =
4
s2 + 6s+ 13
11. Given y(0) = 4, and y˙(0) = 3, use Laplace transforms to solve
y¨ − 3y˙ + 2y = 4t
12. Consider the dynamical system described by
y¨ + 4y˙ + 4y = 4
with y(0) = y˙(0) = 0. Find y(t) using Laplace transforms.
Check your solution to the differential equation using MATLAB.
13. Use Laplace transforms to solve the system
dx
dt
= −y, dy
dt
= x
for t ≥ 0, with x(0) = 1, y(0) = 0.
14
14. Use Laplace transforms to solve the system
dx
dt
= x+ y,
dy
dt
= −x− y
for t ≥ 0, with x(0) = 1, y(0) = 0.
15. A mechanical system consists of three springs, σ1, σ2 and σ3 of negligible mass, equal
natural length and spring constant 1N/m, and two objects, m1 and m2 of mass 1 kg
each.
m1
m2
σ1
σ2
σ3
Assuming that the damping is negligible, and that the departure of m1 from its equi-
librium position is y1(t), and that of m2 is y2(t) at time t. The equations of motion
are given by the following system of second order ordinary differential equations:
y
′′
1 = y2 − 2y1
y
′′
2 = y1 − 2y2
Assume the initial conditions are
y1(0) = 1, y
′
1(0) =
√
3
y2(0) = 1, y
′
2(0) = −
√
3
Use Laplace transforms to solve the system of differential equations subject to the
given initial conditions.
16. An object with mass m receives 11 impulses of strength p at 1 second intervals at t = 0,
t = 1, t = 2, . . ., t = 10. The differential equation describing the motion of this object
is
m
dv
dt
= p
10∑
k=0
δ(t− k)
If the object is initially at rest, find its velocity at time t ≥ 0.
17. Use the convolution theorem to find the Laplace transforms of
(a) f(t) =
∫ t
0
eτ (t− τ) dτ (b) f(t) =
∫ t
0
sin(t− τ)e3τ dτ
(c) f(t) =
∫ t
0
δ(t− τ) sinh(2τ) dτ (d) f(t) =
∫ t
0
(1 + τ)(t− τ)2 dτ
15
18. Solve the integral-differential equation∫ t
0
y(τ) dτ − y′(t) = t
for t ≥ 0, with y(0) = 4.
19. In a series RLC-circuit with resistance R = 2Ω, inductance L = 1H, capacitance
C = 0.25F and voltage v = 50V , the current i at time t is determined by the integral-
differential equation
2i+
di
dt
+ 4
∫ t
0
i(τ) dτ = 50
Solve the integral-differential equation to find the current, if initially there is no current
in the circuit.
20. Consider an RC circuit, with resistance R and capacitance C. The voltage v at time
t is always zero, except from t = a to t = b when v = v0 is a constant. The current i
at any time is given by the differential equation
Ri+
q
C
= v
where q is the charge on the capacitor at any time and
v(t) = v0[u(t− a)− u(t− b)]
Since i =
dq
dt
, the differential equation becomes
Ri+
1
C
∫ t
0
i(τ) dτ = v0[u(t− a)− u(t− b)]
Solve the integral equation for the current.
ADDITIONAL EXERCISES
Kreyszig 10th Edition
Problem Set 6.1 (page 210): Q. 1 - 8, 25 - 45
Problem Set 6.2 (page 216): Q. 1 - 21, 23 - 29
Problem Set 6.3 (page 223): Q. 2 - 27
Problem Set 6.4 (page 230): Q. 3 - 12
Problem Set 6.5 (page 237): Q. 1 - 14, 17 - 25
16
MAST20029 Engineering Mathematics
Sheet 4: Sequences and Series
1. (Revision) Use standard limits to determine whether the following sequences are con-
vergent or divergent, and find the limit if it exists.
(a) an =
6
4n
(b) an =
(
n+ 2
n
)n
(c) an =
loge(n
2)
n
(d) an =
loge n
5n
2. (Revision) Which of the following sequences converge? Find their limits if they con-
verge. For part (a) and (b), verify your answer using MATLAB.
(a) an =
n2
3n2 + 4n+ 1
(b) an =
2n2 + 6n+ 2
3n3 − n2 − n
(c) an = n
(−1)n (d) an = cos(npi)
3. (Revision) Decide which of the following sequences converge. Find their limits if they
do.
(a) an =
nn
(n+ 3)n+1
(b) an =
(
1− 1
n2
)n
(c) an =
loge(n+ 1)
n
(d) an =
n(n+ 1)n+1
(n+ 2)n+2
4. (Revision) Find the limit (if it exists) of each of the sequences whose nth term is given
below.
(a)
5n − 9n!
n+ 2n + n!
(b) exp
(
n2 − 2
2n2 + 1
)
(c)
en
3n+ 4
(d) n sin
(pi
n
)
5. Let an =
3
4n
, n ≥ 1.
(a) Is the sequence {an} convergent?
(b) Calculate the first four partial sums of the sequence {an}
(c) Show that the series
∞∑
n=1
an converges and find the sum.
17
6. Let an =
n− 1
n
, n ≥ 1.
(a) Is the sequence {an} convergent?
(b) Calculate the first four partial sums of the sequence {an}
(c) Show that the series
∞∑
n=1
an diverges.
7. In this question, we determine the sum of a telescoping series.
Let an =
1
n(n+ 1)
, n ≥ 1.
(a) Write an in terms of partial fractions.
(b) Using part (a), calculate the nth partial sum sn of the sequence {an}.
(c) Show that the series
∞∑
n=1
an converges and find the sum.
8. Use the integral test to determine whether the following series converge or diverge.
(a)
∞∑
n=1
1
n2 + 4
(b)
∞∑
n=2
1√
n− 1
(c)
∞∑
n=1
1
(n+ 1)2
(d)
∞∑
n=1
2
2n− 1
9. Use the comparison test to determine whether the following series converge or diverge.
(a)
∞∑
n=1
2n
n2 + 1
(b)
∞∑
n=1
n
(n+ 1)(n2 + 3)
(c)
∞∑
n=2
1
n3 − 1 (d)
∞∑
n=1
10n
n+ 42n
10. Use the ratio test to determine whether the following series converge or diverge.
(a)
∞∑
n=1
n!
nn
(b)
∞∑
n=1
(3n)!
(n!)3
(c)
∞∑
n=1
2n
n+ 1
(d)
∞∑
n=1
10n
n+ 42n
18
11. Use the alternating series test to determine whether the following series converge or
diverge.
(a)
∞∑
n=1
(−1)n2n (b)
∞∑
n=1
(−1)n
n1/3
(c)
∞∑
n=1
(−1)nn
n2 + 1
(d)
∞∑
n=1
(−1)n+13n
4n2
12. Determine whether the following series converge or diverge.
(a)
∞∑
n=0
1
4n
(b)
∞∑
n=1
2−n3n−1
(c)
∞∑
n=1
4n
7n+ 1
(d)
∞∑
n=3
(−1)n+1 loge n
n
(e)
∞∑
n=1
√
n
n2 + 2
(f)
∞∑
n=1
sin2 n
n2
(g)
∞∑
n=1
2n
n3
(h)
∞∑
n=1
en√
n!
13. Find (i) the radius of convergence and (ii) the interval of convergence for each of the
following power series.
(a)
∞∑
n=0
2nxn
n+ 1
(b)
∞∑
n=1
xn
loge(n+ 1)
(c)
∞∑
n=1
(3x+ 6)n
n!
(d)
∞∑
n=0
(−1)n(x+ 1)n
(n+ 1)2
14. For each of the following functions, write down the Taylor polynomial for the given
values of a and n and give an expression for the remainder Rn(x).
Also show how you would use the MATLAB command taylor to verify your answer.
(a) f(x) = sinx a = pi/2 n = 3
(b) f(x) =
√
x a = 4 n = 3
(c) f(x) = xex a = −1 n = 4
19
15. Consider the function
f(x) = x sinx, x ∈ R
(a) Find the quadratic Maclaurin polynomial for f .
(b) Write down an expression for the error in terms of x. In what interval does the
unknown constant c lie?
(c) Check your answer for the Maclaurin polynomial and derivative function needed
to calculate the error, using MATLAB.
(d) Use the MATLAB command taylortool to compare f with the Maclaurin poly-
nomial for f of varying degrees.
16. Consider the function
f(x) = sinh x, x ∈ R
(a) Determine the cubic Maclaurin polynomial for f .
(b) Determine an upper bound for the error on the interval |x| ≤ 1.
17. For what values of x does the cubic Maclaurin polynomial for cosx have an error of
(a) less than .01 (b) less than .01 of |x|?
18. Consider the function
f(x) =
√
1 + x, x ≥ −1
(a) Determine the linear Maclaurin polynomial for f .
(b) Give an expression for the error in terms of x when |x| < 0.19.
(c) Estimate
√
1.1 and determine the size of the error.
19. Consider the function
f(x) =
1
1− x, x 6= 1
(a) By finding the Maclaurin series for f , show that
f(x) =
∞∑
n=0
xn
(b) Find the radius of convergence and interval of convergence for the series in part
(a).
20
20. Consider the function
f(x) = loge(1 + x), x > −1
(a) Find the Maclaurin series for f .
(b) Find the radius of convergence and interval of convergence for the series in part
(a).
(c) Use your Maclaurin series to find value of the alternating series
1− 1
2
+
1
3
− 1
4
+ · · ·
ADDITIONAL EXERCISES
For additional questions on sequences and series look in any first year Calculus textbook
available in the ERC library.
21
MAST20029 Engineering Mathematics
Sheet 5: Fourier Series and Fourier Integrals
1. Consider the waveform
f(t) = et, −pi < t < pi
where f(t) = f(t+ 2pi).
(a) Sketch the graph of f(t) for −3pi ≤ t ≤ 3pi.
(b) Determine the n = 0, 1, 2, 3 terms for the Fourier series of f .
(c) Write down a general expression for the Fourier series of f .
2. A sinusoidal voltage sin t is passed through a half-wave rectifier to produce a periodic
output voltage with period 2pi such that
E(t) =
{
0, −pi < t < 0
sin t, 0 < t < pi
(a) Sketch the graph of E(t) for −3pi ≤ t ≤ 3pi.
(b) Determine the n = 0, 1, 2, 3, 4 terms for the Fourier series of E.
(c) Use Parseval’s Identity to find the first five non zero terms of the energy density
of E.
(d) Calculate the energy density of E.
3. Consider the function
f(t) =
{
1, 0 < t < 1
2, 1 < t < 2
(a) Sketch fe(t), the even periodic extension of f in the range −4 ≤ t ≤ 4.
(b) Determine the first four non zero terms of the Fourier cosine series for fe.
(c) What values does the Fourier series for fe converge to if t = 0 and t = 2?
(d) Sketch fo(t), the odd periodic extension of f in the range −4 ≤ t ≤ 4.
(e) Determine the first four non zero terms of the Fourier sine series for fo.
(f) What values does the Fourier series for fo converge to if t = 0 and t = 2?
4. Consider the function
f(t) = t2 for 0 < t < pi
(a) Sketch fe(t), the even periodic extension of f in the range −3pi ≤ t ≤ 3pi.
(b) Determine the first four non zero terms of the Fourier cosine series for fe.
(c) Sketch fo(t), the odd periodic extension of f in the range −3pi ≤ t ≤ 3pi.
(d) Determine the first four non zero terms of the Fourier sine series for fo.
22
5. Consider the function
f(t) =
{ −(pi + t), −pi < t < 0
pi − t, 0 < t < pi
(a) Obtain a general Fourier series representation for f , if f is periodic with period
2pi.
(b) Using part (a), find a particular solution for the non-homogenous ordinary differ-
ential equation
−y′′ + y = f
(c) Find the general solution for the differential equation in part (b).
6. Consider the function
f(t) =
t2
4
for − pi < t < pi.
(a) Obtain a general Fourier series representation for f , if f is periodic with period
2pi.
(b) Using part (a), find a particular solution to the differential equation
y′′ + y′ + y = f
that describes the oscillations of a damped mechanical system subject to the
periodic driving force f .
(c) Using part (a), evaluate the sum
∞∑
n=1
(−1)n
n2
7. Consider the function
f(t) =

0, t < 0
pi/2, t = 0
pie−t, t > 0
(a) Sketch f(t) for all real values of t.
(b) Obtain the Fourier integral representation for f .
23
8. Consider the function
f(t) =

1, 0 ≤ t ≤ 1
2, 1 < t ≤ 4
0, t > 4
(a) Sketch f(t).
(b) Obtain the Fourier sine integral representation for f .
(c) Obtain the Fourier cosine integral representation for f .
(d) What values do the Fourier integrals in (b) and (c) converge to if t = 1?
(e) What values do the Fourier integrals in (b) and (c) converge to if t = 4?
(f) What values do the Fourier integrals in (b) and (c) converge to if t = 0?
9. Consider the function
f(t) =

pi
2
cos t, 0 ≤ t < pi/2
0, t ≥ pi/2
(a) Sketch fe(t), the even extension of f , for all real values of t.
(b) Obtain the Fourier cosine integral representation for fe.
10. Consider the function
f(t) = −e−2t for t > 0
(a) Sketch fo(t), the odd extension of f , for all real values of t.
(b) Obtain the Fourier sine integral representation for fo.
11. Consider the inhomogeneous differential equation
y′′ + y = f
where
f(t) =
{
sin t, |t| ≤ pi
0, |t| > pi
(a) Find the Fourier integral representation for f .
(b) Using part (a), find a particular solution to the differential equation.
24
12. Consider an infinitely long beam on an elastic foundation with deflection u satisfying
the nonhomogeneous differential equation
EI
d4u
dx4
+ ku = y
where the constant k is the “foundation modulus” (that is, the spring stiffness per
unit x length) and EI the “flexural rigidity” of the beam and y is a prescribed loading
(force per unit x length) given by
y(x) =

ω0, |x| < 1
ω0/2, |x| = 1
0, |x| > 1
(a) Find the Fourier integral representation for y.
(b) Using part (a), find a particular solution to the differential equation.
ADDITIONAL EXERCISES
Kreyszig 10th Edition
Problem Set 11.1 (page 482): Q. 6 - 10, 12 - 21
Problem Set 11.2 (page 490): Q. 8 - 17
Problem Set 11.7 (page 517): Q. 7, 8, 16,17
25
MAST20029 Engineering Mathematics
Sheet 6: Second Order Partial Differential Equations
1. State the order and the number of independent variables for the following PDE’s. De-
cide whether the equation is linear or nonlinear, and homogeneous or inhomogeneous.
(a) φ
∂2φ
∂x2
+
∂φ
∂y
= 1 (b) φ2
∂2φ
∂x2
+
∂2φ
∂y∂z
= 1
(c)
∂4φ
∂x4
+
∂2φ
∂y2
= 0 (d)
∂2φ
∂x2
+
∂3φ
∂y2∂z
= φ2
2. Consider the second order wave equation
∂2φ
∂t2
= 36
∂2φ
∂x2
for 0 < x < pi and t > 0, subject to the boundary and initial conditions
φ(0, t) = 0
φ(pi, t) = 0
φ(x, 0) = sin(2x)
∂φ
∂t
(x, 0) = sinx
(a) Using the method of separation of variables, show that the wave equation reduces
to two ordinary differential equations (ODE’s) of the form:
X ′′(x)− λX(x) = 0
T ′′(t)− 36λT (t) = 0
where λ is the separation constant.
(b) By solving the ODE’s in part (a) for the case λ < 0, determine the solution of
the wave equation subject to the given initial and boundary conditions.
You may assume that solving the ODE’s in part (a) for the cases λ > 0 and λ = 0
leads to trivial solutions. You do not need to work through these two cases.
26
3. Using separation of variables, solve the wave equation
∂2φ
∂x2
− ∂
2φ
∂t2
= 0
for {(x, t) : 0 < x < 1, t > 0}, subject to the boundary conditions
φ(0, t) = 0
φ(1, t) = 0
∂φ
∂t
(x, 0) = 0
φ(x, 0) = sin(2pix)
4. Solve Laplace’s equation
∂2φ
∂x2
+
∂2φ
∂y2
= 0
for {(x, y) : 0 < x < L, 0 < y < L}, subject to the boundary conditions
φ(x, 0) = 0
∂φ
∂x
(0, y) = 0
∂φ
∂x
(L, y) = 0
φ(x, L) = EL+ cos
(pix
L
)
where E is a constant.
5. Solve Laplace’s equation
∂2φ
∂x2
+
∂2φ
∂y2
= 0
in the region
{(x, y) : 0 < x < pi, 0 < y < 1}
subject to the boundary conditions
φ(x, y) =

0 at x = 0
0 at x = pi
sinx+ sin(2x) at y = 0
2 sinx at y = 1
27
6. Solve Laplace’s equation
∂2φ
∂x2
+
∂2φ
∂y2
= 0
in the region
{(x, y) : 0 < x < 1, 0 < y < 1}
subject to the boundary conditions
φ(x, y) =

0 at x = 0
0 at x = 1
sin(pix) at y = 0
sin(2pix) at y = 1
7. Consider the heat equation
∂φ
∂t
= 4
∂2φ
∂x2
for {(x, t) : 0 < x < 1, t > 0}, subject to the boundary and initial conditions
φ(0, t) = 0
φ(1, t) = 0
φ(x, 0) = sin(3pix)
(a) Using the method of separation of variables, show that the heat equation reduces
to two ordinary differential equations (ODE’s) of the form:
X ′′(x)− λX(x) = 0
T ′(t)− 4λT (t) = 0
where λ is the separation constant.
(b) By solving the ODE’s in part (a) for the case λ < 0, determine the solution of
the heat equation subject to the given initial and boundary conditions.
You may assume that solving the ODE’s in part (a) for the cases λ > 0 and λ = 0
leads to trivial solutions. You do not need to work through these two cases.
28
8. (a) Determine the Fourier sine series for the function
f(x) =
{
1 0 < x ≤ L
2
0 L
2
< x ≤ L
(b) Using your answer to part (a), solve the diffusion equation
∂φ
∂t
=
∂2φ
∂x2
for {(x, t) : 0 < x < L, t > 0} subject to the boundary conditions
φ(0, t) = 0
φ(L, t) = 0
φ(x, 0) = f(x)
9. (a) Determine the Fourier cosine series for the function
g(x) = x(L− x), 0 < x < L
(b) Using your answer to part (a), solve the diffusion equation
∂φ
∂t
=
∂2φ
∂x2
for {(x, t) : 0 < x < L, t > 0} subject to the boundary conditions
∂φ
∂x
(0, t) = 0
∂φ
∂x
(L, t) = 0
φ(x, 0) = g(x)
ADDITIONAL EXERCISES
Kreyszig 10th Edition
Problem Set 12.1 (page 542): Q. 2 - 13
Problem Set 12.3 (page 551): Q. 5 - 14
29
MAST20029 Engineering Mathematics
Sheet 1 Numerical Answers
1. (b) ∇ · v = 0, ∇× v = 4k
2. (b) ∇ · v = 2xy − 4y, ∇× v = −x2k
3. (a) 0
(b) 3
4. (a) 0
(b) (10y − 8z)i− (10x− 6z)j + (8x− 6y)k
5. (a)
45
4
(b) 9(e− 1)
6. (a)
9
2
(b) piab
7. (a)(ii)
∫ 1
0
∫ x
x2
dydx
(iii)
1
6
(iv)
>> syms x y
>> int(int(1,’y’,x^2,x),’x’,0,1)
(b)(ii)
∫ 2
−2
∫ √4−x2
0
y dydx
(iii)
16
3
(iv)
>> syms x y
>> int(int(y,’y’,0,sqrt(4-x^2)),’x’,-2,2)
(c)(ii)
∫ 4
0
∫ √x
0
yex
2
dydx
(iii)
1
4
(
e16 − 1)
(iv)
>> syms x y
>> int(int(y*exp(x^2),’y’,0,sqrt(x)),’x’,0,4)
30
(d)(ii)
∫ e
1
∫ 1
loge y
dxdy
(iii) e− 2
(iv)
>> syms x y
>> int(int(1,’x’,log(y),1),’y’,1,exp(1))
8. (a)
0 ≤ r ≤ 2
0 ≤ θ ≤ 2pi
(b) 24pi
9. (a)
1 ≤ r ≤ 2
0 ≤ θ ≤ 2pi
(b) 30pi
10. (a) pi[1− cos(4)]
(b)
pi
4
(e4 − 1)
11. (a)
0 ≤ z ≤ 3− 3x− 3y
0 ≤ y ≤ 1− x
0 ≤ x ≤ 1
(b)
1
2
12. (a)
0 ≤ z ≤ 6− 6x− 3y
0 ≤ y ≤ 2− 2x
0 ≤ x ≤ 1
(b) 2
13. (a)
0 ≤ z ≤ 4− r2
0 ≤ r ≤ 2
0 ≤ θ ≤ 2pi
(b) 0
(c) >> syms r t z
>> int(int(int(r^2*sin(t),’z’, 0, 4-r^2),’r’,0,2), ’t’, 0, 2*pi)
14. (a) 0 ≤ z ≤ 4
0 ≤ r ≤ 2
−pi/2 ≤ θ ≤ pi/2 OR 0 ≤ θ ≤ pi/2, 3pi/2 ≤ θ ≤ 2pi
(b)
128
3
31
15. (b)
pi
4
16. (a) 0 ≤ r ≤ a
0 ≤ φ ≤ pi/2
0 ≤ θ ≤ 2pi
(b)
2pia3
3
17. (a) 0 ≤ r ≤ 2
0 ≤ θ ≤ pi
0 ≤ φ ≤ pi
(b)
128pi
15
18. (a) 0 ≤ r ≤ 1
0 ≤ θ ≤ 2pi
0 ≤ φ ≤ pi
3
(b)
pi
3
(c) >> syms r t p
>> int(int(int(r^2*sin(p),’r’,0,1),’t’,0,2*pi),’p’,0,pi/3)
19. Many answers are possible for each parametrisation.
(a) x(t) = −1 + 4t, y(t) = 2 + t for 0 ≤ t ≤ 1
(b) x(t) = 2 cos t, y(t) = 2 sin t for 0 ≤ t ≤ pi
2
(c) x(t) = 2 cos t, y(t) = −2 sin t for pi ≤ t ≤ 3pi
2
(d) x(t) = 2 cos t, y(t) = −3 sin t for pi ≤ t ≤ 2pi
(e) x(t) = t, y(t) = 2t− t2 for 0 ≤ t ≤ 2
(f) x(t) = 2− t, y(t) = 2(2− t)− (2− t)2 = 2t− t2 for 0 ≤ t ≤ 2
20. (a) 27
(b) >> t=[0:.01:pi/2];
>> x = 3*sin(t);
>> y = 3*cos(t);
>> plot(x,y)
32
21. (a) x(t) = t, y(t) = t3/2 for 1 ≤ t ≤ 4
(b)
1
27
(403/2 − 133/2)
(c) >> t=[1:.01:4];
>> x = t;
>> y = t.^(3/2);
>> plot(x,y)
22. (b)
√
5
(
3pi2 + 2pi
)
23.
195
4
24. (a)
53
20
(b) 3
(c)
7
20
25. (a) yes, φ = x2 + 3
2
y2 + 2z2 + c
(b) no
(c) yes, φ = x2yez + 1
3
z3 + c
26. (a) φ = x2yz − cosx+ c, 38− cos(3) + cos(1)
(b) φ = xz3 + x2y + c, 202
27. (a) e sin(1) +
1
3
e3 − 1
3
(b) −3
2
28.
(653/2 − 1)pi
6
29. 16
√
2pi
30.
9
√
6
8
31. 0
32.
15pi
2
33.
4pi
3
34.
3pi
2
33
35. 60
36. 8pi
37.
8pia3
3
38. 0
39. 6
40. −2pi
41. 0
42. 2pi
43. (b) 16pi
34
MAST20029 Engineering Mathematics
Sheet 2 Numerical Answers
1. (a) eigenvalues 1,−1 and eigenvectors
(
1
1
)
,
(
3
5
)
>>m=[4 -3; 5 -4];
>> [t,e]=eig(m)
t =
[ 0.7071, 0.5145]
[ 0.7071, 0.8575]
e =
[ 1, 0]
[ 0, -1]
(b)
(
x(t)
y(t)
)
= α1
(
1
1
)
et + α2
(
3
5
)
e−t
2. (a)
(
x(t)
y(t)
)
= α1
(
cos t
sin t
)
et + α2
(
sin t
− cos t
)
et
3. (a)
(
x(t)
y(t)
)
= α1
(
3
−1
)
e−2t + α2
(
3t+ 1
−t
)
e−2t
(b)
(
x(t)
y(t)
)
= α1
(
1
1
)
e2t + α2
(
t− 1
t
)
e2t
4. (a) x(t) = −1
5
e−3t +
6
5
e2t, y(t) =
4
5
e−3t +
6
5
e2t
>> [x,y]=dsolve(’Dx=x+y’,’Dy=4*x-2*y’,’x(0)=1’,’y(0)=2’)
(b) x(t) = 2et cos t+ 6et sin t, y(t) = −et sin t+ 3et cos t
>> [x,y]=dsolve(’Dx=x+2*y’,’Dy=-0.5*x+y’,’x(0)=2’,’y(0)=3’)
35
5. (a) (i)
(ii) The origin is an unstable spiral, oriented anticlockwise.
(b) (i)
(ii) The origin is an unstable node.
(c) (i)
(ii) The origin is an unstable saddle.
36
(d) (i)
(ii) The origin is an asymptotically stable node.
6. (a) (i)
(
x(t)
y(t)
)
= α1
(
cos t
sin t
)
+ α2
(
sin t
− cos t
)
(ii) The orbits are circles, oriented anticlockwise.
(iii) The origin is a stable centre.
(iv)
(
x(t)
y(t)
)
= 2
(
cos t
sin t
)
− 7
(
sin t
− cos t
)
>> ezplot(’2*cos(t)-7*sin(t)’,’2*sin(t)+7*cos(t)’,([-10,10]))
>> axis([-10 10 -10 10])
(b) (i)
(
x(t)
y(t)
)
= α1
(
1
−3
)
e−7t + α2
(
1
−5
)
e−5t
(ii)
(iii) The origin is an asymptotically stable node.
(iv)
(
x(t)
y(t)
)
=
17
2
(
1
−3
)
e−7t − 13
2
(
1
−5
)
e−5t
>> ezplot(’17*exp(-7*t)/2-13*exp(-5*t)/2’,’-51*exp(-7*t)/2
+65*exp(-5*t)/2’,([-10 10]))
>> axis([-3 3 -3 3])
37
7. (a) (i)
(
x(t)
y(t)
)
= α1
(
cos(2t)
sin(2t)− cos(2t)
)
e−4t + α2
(
sin(2t)
−(sin(2t) + cos(2t))
)
e−4t
(ii) The orbits are anticlockwise spirals.
(iii) The origin is asymptotically stable.
(iv)
(
x(t)
y(t)
)
=
( −2 sin(2t)
2 sin(2t) + 2 cos(2t)
)
e−4t
(b) (i)
(
x(t)
y(t)
)
= α1
(
1
0
)
e−2t + α2
(
0
1
)
e−2t
(ii)
(iii) The origin is an asymptotically stable star node.
(iv)
(
x(t)
y(t)
)
= 2
(
0
1
)
e−2t
8. (a) (0, 0) and (4, 0).
(b) can use linear system to approximate non-linear system near (0, 0) and (4, 0).
(c) (0,0) is an unstable spiral. (4,0) is an unstable saddle.
(d)
38
9. (a) (npi, 0) where n = 0,±1,±2 . . .
(b) can use linear system to approximate non-linear system near (npi, 0) where n =
0,±1,±2 . . .
(c) Critical points at (0, 0) and (npi, 0) for n even, are asymptotically stable spirals.
Critical points at (npi, 0) for n odd, are unstable saddles.
(d)
10. (a) (0, 0)
(b) can use linear system to approximate non-linear system near (0, 0)
(c) The origin is an unstable spiral.
(d)
11. (a) (0, 0), (
√
2, 0), (−√2, 0)
(b) can use linear system to approximate non-linear system near (0,0) only.
(c) 0,0) is an unstable saddle.
(d)
39
MAST20029 Engineering Mathematics
Sheet 3 Numerical Answers
1. (a)
k
s2c
(1− e−sc − cse−sc)
(b)
k
s
(e−sa − e−sb)
(c)
a
s2 − a2
(d)
s
s2 + a2
2. (a)
4
s3
+
6
s
− 3
s− 4
(b)
Aω
s2 + ω2
+
Bs
s2 − k2
(c)
24
s5
+
120
s6
(d)
s2 + 2
s(s2 + 4)
>> syms t w A B k
>> laplace(2*t^2+6-3*exp(4*t))
>> laplace(A*sin(w*t)+B*cosh(k*t))
>> laplace(t^4+t^5)
>> laplace((cos(t))^2)
3. (a) 3t
(b) sin(4t)
(c) cos(3t)
(d) 5 cosh(5t)
>> syms s
>> ilaplace(3/s^2)
>> ilaplace(4/(s^2+16))
>> ilaplace(s/(s^2+9))
>> ilaplace(5*s/(s^2-25))
4. (a) 1
5
(4e−4t + et)
(b) 1
3
(e2t − e−t)
40
5. (a)
3
(s− 2)2
(b)
20
(s+ 6)2 + 16
(c)
6
(s+ 1)2 − 9
(d)
s− 1
(s− 1)2 + 1 −
s+ 1
(s+ 1)2 + 1
6. (a) 3te−2t
(b)
t2e2t
2
(c) e4t sin t
(d) e−2t cos(2t)
7. (a)
se−s
s2 − 1
(b)
e−pis
s2
+
pie−pis
s
>> syms t
>> laplace(t*heaviside(t-pi))
8. (a)
1− e2−2s
s− 1
(b)
s
s2 + pi2
(−e−s − e−4s)
(c)
1
s
+
(
1
s2
+
4
s
)
e−5s −
(
1
s2
+
10
s
)
e−10s
(d)
−pi
s2 + pi2
(
e−s + e−2s
)
+
e−6−3s
s+ 2
9. (a) u(t− 3)
(b) 5u(t− 2) + 4(t− 2)u(t− 2)
10. (a) u(t− 3)e−8(t−3)
(b)
1
5
u(t− 2)(e4(t−2) − e−(t−2))
(c) e−2t[cos(2t) + 2 sin(2t)]
(d)
1
2
[1− cos(2t)]
(e) 2δ(t) +
1
4
sinh[4(t− 1)]u(t− 1)
(f) 2 sin(2t)e−3t
11. y(t) = 3 + 2t+ et
41
12. y(t) = 1− e−2t − 2te−2t
>> y=dsolve(’D2y+4*Dy+4*y=4’,’y(0)=0’,’Dy(0)=0’)
13. x(t) = cos t, y(t) = sin t
14. x(t) = 1 + t, y(t) = −t
15. y1(t) = cos t+ sin(
√
3t), y2(t) = cos t− sin(
√
3t)
16. v(t) =
p
m
10∑
k=0
u(t− k)
17. (a)
1
s2(s− 1)
(b)
1
(s− 3)(s2 + 1)
(c)
2
s2 − 4
(d)
2
s4
+
2
s5
18. y(t) = 1 +
3
2
(et + e−t)
19. i(t) =
50√
3
e−t sin(
√
3t)
20. i(t) =
v0
R
[
e−(t−a)/(RC)u(t− a)− e−(t−b)/(RC)u(t− b)]
42
MAST20029 Engineering Mathematics
Sheet 4 Numerical Answers
1. (a) 0
(b) e2
(c) 0
(d) 0
2. (a)
1
3
>> syms n
>> an = n^2/(3*n^2+4*n+1)
>> limit(an,n,inf)
(b) 0
>> syms n
>> an = (2*n^2+6*n+2)/(3*n^3-n^2-n)
>> limit(an,n,inf)
(c) divergent
(d) divergent
3. (a) 0
(b) 1
(c) 0
(d)
1
e
4. (a) −9
(b)
√
e
(c) divergent
(d) pi
5. (a) converges to 0
(b)
3
4
,
15
16
,
63
64
,
255
256
(c) geometric series |r| < 1. Sum is 1.
43
6. (a) converges to 1
(b) 0,
1
2
,
7
6
,
23
12
(c) Divergent as terms an do not approach 0.
7. (a) an =
1
n
− 1
n+ 1
(b) sn = 1− 1
n+ 1
(c) s = 1
8. (a) convergent
(b) divergent
(c) convergent
(d) divergent
9. (a) divergent
(b) convergent
(c) convergent
(d) convergent
10. (a) convergent
(b) divergent
(c) divergent
(d) convergent
11. (a) divergent
(b) convergent
(c) convergent
(d) divergent
12. (a) convergent
(b) divergent
(c) divergent
(d) convergent
(e) convergent
(f) convergent
(g) divergent
(h) convergent
44
13. (a) (i)R =
1
2
(ii) −1
2
≤ x < 1
2
(b) (i) R = 1 (ii) −1 ≤ x < 1
(c) (i) R =∞ (ii) x ∈ R
(d) (i) R = 1 (ii) −2 ≤ x ≤ 0
14. (a) P3(x) = 1− 1
2
(x− pi/2)2
R3(x) =
sin(c)(x− pi/2)4
24
where c lies between pi/2 and x
>> syms x
>> f= sin(x)
>> taylor(f,x,pi/2,’Order’,4)
(b) P3(x) = 2 +
1
4
(x− 4)− 1
64
(x− 4)2 + 1
512
(x− 4)3
R3(x) =
−5(x− 4)4
128c
7
2
where c lies between 4 and x
>> syms x
>> f= sqrt(x)
>> taylor(f,x,4,’Order’,4)
(c) P4(x) =
−1
e
+
(x+ 1)2
2e
+
(x+ 1)3
3e
+
(x+ 1)4
8e
R4(x) =
(5 + c)ec
120
(x+ 1)5 where c lies between −1 and x
>> syms x
>> f= x*exp(x)
>> taylor(f,x,-1,’Order’,5)
15. (a) P2(x) = x
2
(b) |R2(x)| = 16 |3 sin(c) + c cos(c)|x3 where c lies between 0 and x
(c) >> syms x
>> f=x*sin(x)
>> taylor(f,x,0,’Order’,3)
>> diff(f,3)
>> taylortool(’x*sin(x)’)
16. (a) P3(x) = x+
x3
6
(b) |R3| < sinh(1)
24
<
1
16
17. (a) |x| < 4
√
6
25
(b) |x| < 3
√
6
25
45
18. (a) 1 +
x
2
(b) |R1(x)| < x
2
8(0.81)1.5
<
x2
5
(c)
√
1.1 ≈ 1.05, |R1| = 1
800(1 + c)1.5
<
1
800
19. (b) R = 1, −1 < x < 1
20. (a) f(x) =
∞∑
n=1
(−1)n+1 xn
n
(b) R = 1, −1 < x ≤ 1
(c) loge(2)
46
MAST20029 Engineering Mathematics
Sheet 5 Numerical Answers
1. (b) f(t) =
1
pi
(epi − e−pi)(1
2
− 1
2
cos t+
1
5
cos(2t)− 1
10
cos(3t) + . . .
+
1
2
sin t− 2
5
sin(2t) +
3
10
sin(3t) + . . .)
(c) f(t) =
1
pi
(epi − e−pi)
{
1
2
+
∞∑
n=1
(−1)n
1 + n2
[cos(nt)− n sin(nt)]
}
2. (b) E(t) =
1
pi
+
1
2
sin t−
(
2
3pi
)
cos(2t)−
(
2
15pi
)
cos(4t)− . . .
(c)
1
pi2
+
1
2
[(
1
2
)2
+
(
2
3pi
)2
+
(
2
15pi
)2
+
(
2
35pi
)2
+ . . .
]
(d)
1
4
3. (b) fe(t) =
3
2
− 2
pi
cos
(
pit
2
)
+
2
3pi
cos
(
3pit
2
)
− 2
5pi
cos
(
5pit
2
)
+ . . .
(c) 1, 2 respectively
(e) fo(t) =
6
pi
sin
(
pit
2
)
− 2
pi
sin (pit) +
6
3pi
sin
(
3pit
2
)
+
6
5pi
sin
(
5pit
2
)
+ . . .
(f) 0, 0 respectively
4. (b) fe(t) =
pi2
3
− 4 cos t+ cos(2t)− 4
9
cos(3t) + . . .
(d) fo(t) =
(
2pi2 − 8
pi
)
sin t− pi sin(2t) +
(
18pi2 − 8
27pi
)
sin(3t)− pi
2
sin(4t) + . . .
5. (a) f(t) = 2
∞∑
n=1
sin(nt)
n
(b) yp(t) = 2
∞∑
n=1
sin(nt)
n(n2 + 1)
(c) y(t) = Aet +Be−t + 2
∞∑
n=1
sin(nt)
n(n2 + 1)
6. (a) f(t) =
pi2
12
+
∞∑
n=1
(−1)n
n2
cos(nt)
(b) yp(t) =
pi2
12
− 3
52
cos(2t) + · · · − sin(t) + 1
26
sin(2t)− . . .
(c) −pi
2
12
47
7. (b) f(t) =
∫ ∞
0
cos(ωt) + ω sin(ωt)
1 + ω2
dω
8. (b) fo(t) =
∫ ∞
0
2
ωpi
[1 + cosω − 2 cos(4ω)] sin(ωt) dω
(c) fe(t) =
∫ ∞
0
2
ωpi
[2 sin(4ω)− sinω] cos(ωt) dω
(d) 3
2
(e) 1
(f) 0, 1 respectively
9. (b) fe(t) =
∫ ∞
0
cos(piω/2) cos(ωt)
1− ω2 dω
10. (b) fo(t) =
−2
pi
∫ ∞
0
ω sin(ωt)
ω2 + 4
dω
11. (a) f(t) =
2
pi
∫ ∞
0
sin(ωpi)
1− ω2 sin(ωt) dω
(b) yp(t) =
1
pi
∫ ∞
0
2 sin(ωpi) sin(ωt)
(1− ω2)2 dω
12. (a) y(x) =
2ω0
pi
∫ ∞
0
sinω
ω
cos(ωx) dω
(b) yp(x) =
∫ ∞
0
2ω0
piω
sinω
(EIω4 + k)
cos(ωx) dω
48
MAST20029 Engineering Mathematics
Sheet 6 Numerical Answers
1. (a) Second order, 2 variables, nonlinear, inhomogeneous.
(b) Second order, 3 variables, nonlinear, inhomogeneous.
(c) Fourth order, 2 variables, linear, homogenous.
(d) Third order, 3 variables, nonlinear, homogenous.
2. (b) φ(x, t) = sin(2x) cos(12t) +
1
6
sinx sin(6t)
3. φ(x, t) = sin(2pix) cos(2pit)
4. φ(x, y) = Ey +
1
sinh(pi)
cos
(pix
L
)
sinh
(piy
L
)
5. φ(x, y) = sin x[α1 sinh y + cosh y] + sin(2x)[α2 sinh(2y) + cosh(2y)]
where α1 =
2− cosh(1)
sinh(1)
, α2 =
− cosh(2)
sinh(2)
6. φ(x, y) = sin(pix)[α1 sinh(piy) + cosh(piy)] + α2 sin(2pix) sinh(2piy)
where α1 =
− cosh(pi)
sinh(pi)
and α2 =
1
sinh(2pi)
7. (b) φ(x, t) = sin (3pix) e−36pi
2t
8. (a) f(x) =
∞∑
n=1
2
npi
[1− cos(npi/2)] sin
(npix
L
)
(b) φ(x, t) =
∞∑
n=1
2
npi
[1− cos(npi/2)] sin
(npix
L
)
exp
[
−
(npi
L
)2
t
]
9. (a) g(x) =
L2
6
−
∞∑
n=1
L2
(npi)2
cos
(
2npix
L
)
(b) φ(x, t) =
L2
6
−
∞∑
n=1
L2
(npi)2
cos
(
2npix
L
)
exp
[
−
(
2npi
L
)2
t
]
49