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
Week 6: Lectures A, B and C
Euler Equation
and
Coupled Homogeneous ODEs
Plan for Week 6
(a) Euler Equation
Lecture Notes:Ch3B, 3.3; pages 18-19
(b) Coupled Ordinary Differential Equations
Lecture Note
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
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) Nonhomogene
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.
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
(Ch
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,
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
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 P
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
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)
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 rel
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
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
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 nit
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
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 die
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 tex
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 Equat