MA209
DIFFERENTIAL EQUATIONS
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
MARCH 2015
Chapter 1
Constructing ordinary
dierential equations
1.1
Linear equations
Linear equations are those where the highest power is one,

MA209 Assignment 2 (2015)
Question 2.1: (a) Let A R22 . Does the inverse of eA exist? If yes what is
(eA )1 equal to? Explain.
(b) Let t 0 and
1
te2t
te2t 1
B(t) =
.
For which values of t is the matrix B(t) invertible?
(c) Use parts (a) and (b) to deduce

MA209 Assignment 1 (2015)
Question 1.1: (a) State wether the statement is True or False. Give a brief
explanation. If the statement is True you do not need to prove it. If it is False
give counterexample(s).
A dierential equation x (t) = f (x(t) with x(t

MA209 Assignment 3 (2015)
Question 3.1: (a) Let A R22 and let (A) be the spectrum of A. Explain,
in terms of (A), when the system x (t) = Ax(t),
(i) has only bounded solutions.
(ii) has an unbounded solution.
Examine cases (i) and (ii) separately.
In what

MA209 Round-Table Class 7 (2015)
Learning Objectives
After studying sections 3.1 3.6, practising the exercises in Round Table Class
7 and completing Assignment 7 you will be able:
To understand the concept of an equilibrium point in the context of linear

MA209 Round-Table Class 5 (2015)
Learning Objectives
To revise concepts and methods from sections 2.4 2.5.
To perform a phase plane analysis of 2D-linear systems based on phase
portraits and eigenvalues of the corresponding matrix.
To locally linearize

MA209 Assignment 6 (2015)
Question 6.1: (a) Give and example of two sets A and B in the positive
quadrant of R2 that are separated. You may use any results from the course
without proof.
(b) Let A and B be two subsets of R2 . The distance between A and B

MA209 Assignment 5 (2015)
Question 5.1: In what follows consider the 2D linear system x = Ax where
A=
3 0
0 0
.
(a) Find the equilibrium points of the system.
(b) Draw the phase portrait (without the use of MAPLE).
(c) Show that the point xe = (0, 2) is a

MA209 Assignment 4 (2015)
Question 4.1: (a) Give an example of a 2D system x (t) = f (x(t) such that
each point on the line x2 = 2x1 is an equilibrium point.
(b) Let = 0 and consider the 1D system given by
x = x(4 x2 )
Find all the equilibrium points of t

MA209 Round-Table Class 1 (2015)
Learning Objectives
To revise the main concepts from sections 1.1 1.4.
To classify dierential equations.
To verify that a function is a solution to a given dierential equation.
To transform a high order dierential equa

MA209 Assignment 7 (2015)
Question 7.1: (a) Using the denitions show that exponential stability implies
asymptotic stability.
(b) Consider the 2D nonlinear system given by
x1 = x2 x1 x2 + x2
1
2
2
2
x2 = x1 x2 x1 + x2
i. Find the equilibrium points of the

[SE
Summer 2014 examination
MA209
Differential Equations
Half Unit
Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This paper contains 5 questions. You may attempt as many questions as you wish, but only your BEST
4 answers wi

Summer 2012 examination
MA209
Dierential Equations
Half Unit
Suitable for all candidates
Instructions to candidates
Time allowed: 2 hours
This examination paper contains 5 questions. You may attempt as many questions as
you wish, but only your best 4 answ

MA209 (2015)
Self-Study Exercises
3 0
. Find an initial condition x0
1 2
such that the unique solution to the IVP
Exercise 2: Consider the matrix A =
with 0 < x0
2
<
1
3
x (t) = Ax(t) for t 0 with x(0) = x0 .
is (a) not bounded and (b) bounded. In each ca

MA209 (2015)
Self-Study Exercises
Exercise 1: Show that the compound interest Initial Value Problem
w = r(t)w(t)
with w(0) = w0
has a unique solution given by
w(t) = w0 e
t
0
r(s)ds
.
Solution to Exercise 1: The function w(t) = w0 e
IVP since
(a) w(0) = w

MA209 Round-Table Class 8 (2015)
Learning Objectives
After studying sections 3.7 3.9, practising the exercises in Round Table Class
8 and completing Assignment 8 you will be able:
To state what is a Lyapunov function
To understand and explain the Lyapun

MA209 Round-Table Class 9 (2015)
Learning Objectives
After studying sections 4.1 4.5 in the lecture slides, practising the exercises in
Round Table Class 9 and completing Assignment 9 you will be able:
To know the concept of the Lipschitz condition in di

MA209 Round-Table Class 3 (2015)
Learning Objectives
To revise concepts and methods from sections 1.5 1.7.
To compute the matrix exponential etA using the Laplace transform method
or the Jordan canonical form method.
To nd the matrix A when you know th

MA209 Round-Table Class 6 (2015)
Learning Objectives
After studying sections 2.6A 2.6B, practising the exercises in Round Table
Class 6 and completing Assignment 6 you will be able:
To understand the topological concepts related to the phase plane analys

MA209 Round-Table Class 4 (2015)
Learning Objectives
To revise concepts and methods from sections 2.1 2.3.
To nd the singular points of 1D and 2D-systems.
To draw phase portraits for 1D and 2D-systems using either an analytical
method or MAPLE (instruc

MA209 Round-Table Class 2 (2015)
Learning Objectives
To revise the main concepts from sections 1.5 1.7.
To know the denition and properties of the matrix exponential eA and of
the matrix valued function etA .
To know the denition of the norm, A
involvi

MA209 Assignment 9 (2015)
Question 9.1: Find a Lipschitz constant L > 0, for the function f : R R,
2
dened by f (x) = ex .
Hint: Take x2 > y 2 and observe that
2
2
| ex ey |=| (ex
2 +y 2
2
1)ey | .
Apply the Mean Value theorem on the function ea for a <

MA209 Assignment 8 (2015)
Question 8.1: Consider the 2D system
x1 = x1 + x3
2
x2 = x1 x2
(a) Show that the origin is the only equilibrium point.
(b) For which values of and is V (x1 , x2 ) = x2 + x4 a Lyapunov function
1
2
for the system?
(c) Is the origi