MA1506
Mathematics II
Group A
Lecturer: Chew Tuan Seng
Chew T S MA1506-12 Chapter 1
1
Chapter 1
Ordinary Differential Equations
Chew T S MA1506-12 Chapter 1
2
1.1. Introduction
In this Chapter, we study
1st order ordinary differential equation
and its app

MA1506
1.
Tutorial 11
Solutions
Let S(t) be the number of Spartans, P (t) the number of Persians. We have
dS
= P
dt
dP
= 11, 111, 111.1S
dt
so
dS
dt
dP
dt
[
0
1
=
11, 111, 111.1 0
][
S
P
]
Eigenvalues 2 11, 111, 111.1 = 0 = 3333.3333
[
Eigenvectors
3333.3

CHAPTER 2. OSCILLATIONS
2.1. THE HARMONIC OSCILLATOR
Consider the pendulum shown.
The small object, mass m,
at the end of the pendulum,
is moving on a circle of radius
L, so the component of its velocity
tangential to the circle is L
Hence its tangential

Chapter 8. Partial Differential Equations
8.1
Partial Differential Equations
Mathematical models of physical phenomena often
involve differential equations with more than one independent variable. For instance, a model of heat
conduction in a region (e.g.

MA1506 LECTURE NOTES
CHAPTER 1
DIFFERENTIAL EQUATIONS
1.1
Introduction
A differential equation is an equation that
contains one or more derivatives of a differentiable function. [In this course we deal only with
ordinary DEs, NOT partial DEs.]
The order o

MA1506 TUTORIAL 11
Question 1
King Xerxes I of Persia has sent a million soldiers to conquer Greece. King Leonidas I of
Sparta decides to meet the Persians at Thermopylae. A typical Persian soldier can kill
one Spartan per hour, whereas a typical Spartan

CHAPTER 7
SYSTEMS OF FIRST-ORDER ODEs
7.1. ROMEO AND JULIET
We all know that many (most) relationships
have their ups and downs. Lets try to model
this fact.
Romeo loves Juliet, but Juliet believes in a
more subtle approach and finds Romeos excessive enth

The E=0 case is just the logistic case and
we have already studied that.
In the following we shall consider the
remaining three cases.
Case 1:
Case 2:
Suppose we start with
Case 3:
(For the top curve in the next picture)
(See the next picture)

MA1506 TUTORIAL 10
Question 1
Wherever possible, diagonalize the matrices in question 3 of Tutorial 9. [That is, write
them in the form PDP1 , after nding P and D, where D is diagonal.] Using this, work
out their 4th powers.
[Answers: The rst one cannot b

MA1506 TUTORIAL 4 SOLUTIONS
Question 1
(i) x
= cosh(x). An equilibrium solution of an ODE is just a solution that is identically
constant. That is not possible here because the cosh function never vanishes. So there is
no equilibrium for this ODE.
(ii) x

ANSWERS TO MA1506 TUTORIAL 5
Question 1
Following the standard equations for the Malthus Model [Chapter 3]:
ekt ; N (0) = 10000 = N
N =N
N (2.5) = 10000e2.5k = 11000
1
e2.5k = 1.1 k =
n(1.1)
2.5
= 0.0381
N (10) = 10000e10k = 10000e10(0.0381) 14600
1
200

MA1506
Mathematics II
Group A
Lecturer: Chew Tuan Seng
follow the contents of Lecture Note but my
presentation may be different
Chew T S MA1506-12
Chapter 1
1
Chapter 1
Ordinary Differential Equations
Chew T S MA1506-12
Chapter 1
2
1.1. Introduction
In th

MA1506
Mathematics II
Chapter 2
Oscillations
Chew T S MA1506-12 Chapter 2
1
In Chapter two,
we are interested in properties of solutions
We have learnt how to solve those ODEs
in Chapter one, so we are not interested in
how to solve those ODE here
You don

NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2011-2012
MA1506
Mathematics II
Nov/Dec 2011 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper contains FOUR (4) questions and comprises SIX (6)
printed p

NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER 1 EXAMINATION 2012-2013
MA1506
Mathematics II
November/December 2012 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper contains FOUR (4) questions and comprises

MA1506
Mathematics II
Chapter 6
Linear Transformation
Chew T S MA1506-15 Chapter 6
1
In this chapter
We shall prove that
linear transformation=matrix
determinant=volume
and understand why AX=B has solutions or no
solutions from geometrys point of view
Che

MA1506
Mathematics II
Chapter 3
Mathematical Modelling
Chew T S MA1506 Chapter 3
1
Introduction
mathematical model uses mathematical
language to describe a system.
In this module, we use ODE to describe some
systems.
In the last two chapters, we have stud

MA1506
Mathematics II
Chapter 8
Partial Differential equations
PDEs
Chew T S MA1506 Chapter 8
1
In this Chapter, we will study
(1) Method of separation of variables (MSV)
(2) Wave equation (Two versions)
(a)by MSV (b) by dAlemberts method
(3) Heat equatio

MA1506
Mathematics II
Chapter 7
Systems of First Order ODEs
Chew T S MA1506-15 Chapter 7
1
7.1 Solving Linear System of ODEs
How to solve
dx
= ax + by
dt
i.e,
dy
= cx + dy
dt
a,b,c,d constants
d x a b x
=
y
c
d
y
dt
We shall look at an old problem,
wh

MA1506
Mathematics II
Chapter 5
Matrices and their uses
This chapter consists of two parts
Chew TS MA1506 SEM 1 Chapter 5
1
PART ONE
In part one, we shall study
Matrix operations
Some special matrices
Inverse matrix and unique solution of AX=B
Determinant

MA1506
Mathematics II
Chapter 2
More Applications of ODEs
Chew T S MA1506 Chapter 2
1
In this Chapter, we study
Applications of 2nd order ODEs
and
Cantilevered Beam (4th order ODE)
Chew TS MA1506 SEM 1 Chapter 2
2
In Chapter two, we are interested in
prop

MA1506
Mathematics II
Chapter 4
Laplace Transforms
Chew TS MA1506 SEM 1 Chapter 4
1
4.0 Introduction
4.0 Introduction
In Chapter one, we have learnt how to solve
nonhomogeneous 2nd order linear ODE (with
or without initial conditions) by
Method of undete

MA1506
Mathematics II
Group A
Group B
Mon 800-1000
Wed 800-1000
Wed 1600-1700
Fri 800-900
UT-AUD1
Webcast
UT-AUD2
Lecturer: Chew Tuan Seng
follow the contents of Lecture Note but my
presentation may be different
Chew T S MA1506 Chapter 1
1
In this module,

(A) Basic Laplace transforms
n!
1
L(t ) = n+1
,s > a
=
L (e )
sa
s
n!
ct n
1
L (e t ) =
L(1) =
n +1
( s c)
s
n
at
L(sin t ) = 2
s + 2
s
L(cos t ) = 2
2
s +
sc
L(e cos t ) =
L(e sin t ) =
2
2
( s c)2 + 2
( s c) +
ct
ct
Chew T S MA1506-12 Review of lecture