Chapter 2
First Order Ordinary Dierential
Equations
2.1
Separable and Homogeneous Ordinary Dierential
Equations
A general rst order dierential equation can be written in the form
dy
= F (x, y),
dx
for some function F . If F = F (x) only, i.e., the rst ord
Chapter 6:
Second Order Partial Dierential Equations
6.1
Introduction
Many physical processes are governed by second order partial dierential equations. The
main ones are:
Wave Equation
2 u =
1 2u
c2 t2
or
2u 2u 2u
1 2u
+ 2 + 2 = 2 2.
x2 y
z
c t
- propag
Week 8: Lectures A, B and C
Coupled ODEs
Fourier Analysis
Plan for Week 8
(a) Matrix Methods for Homogeneous Coupled
ODEs
Lecture Notes: Ch3C, Pages 22-28
Textbook: Ch4, Pages 129-141)
(b) Nonhomogeneous Linear Systems of ODEs
Lecture Notes: Chapter 3C; P
Chapter 4: Fourier Series
4.1
Introduction
Fourier analysis is concerned with the representation of complicated periodic functions (signals/waveforms) by a combination of simple sine/cosine functions. Fourier analysis can be
carried out on periodic functi
Week 9: Lectures A, B and C
Fourier Analysis
Plan for Week: Fourier Series
(a) Functions of period 2
(Chapter 11, pages 478-486)
(Lecture Notes: Ch4, pages 2-10)
(b) Functions of general period 2L
(Chapter 11, pages 487-490)
(Lecture Notes: Ch4, pages 10-
Week 2 Lectures
Solution of First Order ODEs
Plan for week (First Order ODES)
Exact Differential Equations
(Ch 2, 2.2 of my notes, pages 4-5)
( 1.4 of textbook; pages 19-22)
Integrating Factor
(Ch 2, 2.3 of my notes, pages 6-7)
( 1.5 of textbook; pages 26
Week 12 Lectures A, B and C
Second Order PDEs
Plan for Week (I): PDES
(a) Finish the 1D wave equation
(Pages 540-548 of the textbook;
Chapter 6, pages 2-7 of lecture notes)
(b) General solution rules
(c) Solve the 1D Heat Equation
(Pages 552-557 of the te
Week 10: Lectures
Fourier Analysis
and
First Order PDEs
Plan for Week: Fourier Series
(a) Fourier Series: Periodic Extension
Chapter 11, pages 490-496
Lecture Notes: Ch4, pages 16-21
(b) First Order PDEs
Lecture Notes: Chapter 5
Functions on a Finite Peri
Week 4: Lectures A, B and C
Inhomogeneous Second Linear Order
ODEs
Plan for Week
2nd order linear inhomogeneous ODES
- Method of Undetermined Coefficients
( 3.2 of lecture notes; pages 6-10)
( 2.7 of textbook; pages 78-83)
- Solution by Variation of Param
Lectures Week 5
Variation of Parameter,
Existence, Uniqueness, Dependence,
Euler-Cauchy Equations
Plan for This Week
(a) Existence, Uniqueness and Dependence
(Ch3B, 3.1, 3.2 of the notes; pages 12-14)
( 1.7 of textbook; pages 37-41
2.6 of textbook; pages
Chapter 1
Introduction
This module is primarily concerned with the subject of ordinary and partial dierential equations and in particular how to solve such equations. Many of the physical laws and relations in
engineering mathematics and general science a
Chapter 3
Higher Order Linear Ordinary
Dierential Equations (Part B)
3.1
Existence and Uniqueness of Solutions
In this section the general theory of homogeneous linear ordinary dierential equations is discussed.
Consider the general second order homogeneo
Chapter 3
Higher Order Linear Ordinary
Dierential Equations (Part A)
A general linear second order ordinary dierential equation can be written in the form
d2 y
dy
+ p(x) + q(x)y(x) = r(x).
(3.1)
2
dx
dx
where p(x), q(x) and r(x) are given functions of x.
Chapter 5:
First Order Partial Dierential Equations
5.1
Introduction
A partial dierential equation (PDE) is an equation involving an unknown function, for example
u, of two or more variables and a nite number of its derivatives.
A rst order partial dieren
Complex Numbers
1
Introduction
Consider the general quadratic equation
ax2 + bx + c = 0,
a, b, c (Real Numbers).
where
(1)
The solution of (1) is given by
x=
b
(b2 4ac)
.
2a
(2)
If b2 4ac then b2 4ac 0 and (b2 4ac) will yield a real number since we are
t
Chapter 3
Higher Order Linear Ordinary
Dierential Equations (Part C)
3.1
3.1.1
Homogeneous Systems of Ordinary Dierential
Equations
Introduction
A homogeneous system of n rst order linear ordinary dierential equations with constant
coecients has the form
Week 3 Lectures
Solution of 2nd Order ODEs
Plan for Week 3
Second Order Linear ODES
- constant coefficients
( 3.1, 3.2 of Lecture notes; pages 1-10)
( 2.1, 2.2, of textbook; pages 45-61)
( 2.7, of textbook; pages 78-83)
Trial solution for Homogeneous Equa
Week 11 Lectures A, B and C
First Order PDEs
&
Second Order PDEs
Plan For Week
First Order Partial Differential Equations
(a) Quasi-Linear Equations (Chapter 5)
Second Order Partial Differential Equations
Lecture Notes- Chapter 6, pages 1-3
Textbook Chapt