The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
28
CHAPTER 2. FIRSTORDER ODES
where we may assume the initial condition x(0) = x0 > 0. Separating variables
and integrating
Z x
Z
dx
=
d.
x)
x0 x(1
0
The integral on the lefthandside can be done using the method of partial
fractions:
1
x(1
x)
a
b
+
x
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2016
Math2351(2016 Spring): Tutorial note 06
Zhang Lian and
Li Haotian
Review:
1. Variation of Parameters
y 00 + p(t)y 0 + q(t)y = g(t)
00
0
y + p(t)y + q(t)y = 0
(1)
(2)
Objective: find the special solution Y (t) of Eq.(1)
First, suppose y1 and y2 form a fun
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2016
Math2351(2016 Spring): Tutorial note 02
Zhang Lian and
Li Haotian
Review:
1. Classification of ODE
One way to classify ODE is by linearity, i.e. linear or nonlinear. The order of ODE is defined by the
degree of the highest derivative that appeared in th
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2016
Math2351(2016 Spring): Tutorial note 03
Zhang Lian and
Li Haotian
Review:
1. Separable equations
For a first order ODE,
dy
= f (x, y), if it can be rewritten into this form:
dx
M (x)dx + N (y)dy = 0,
M (x), N (y) are arbitrary functions, only depend on x
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2016
Math2351(2016 Spring): Tutorial note 08
Zhang Lian and
Li Haotian
Review:
As the problems of our course become more and more difficult and time consuming, but we only have 50
minutes each week, so from now on, first things first, we will mainly focus on
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
(MATH2351)[2013](f)midterm~=08bidkdt^_46035.pdf downloaded by jjychan from http:/petergao.net/ustpastpaper/down.php?course=MATH2351&id=1 at 20151022 11:34:01. Academic use within HKUST only.
(MATH2351)[2013](f)midterm~=08bidkdt^_46035.pdf downloaded by
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
4.3. HEAVISIDE AND DIRAC DELTA FUNCTIONS
55
x
x=f(t)
t
Figure 4.1: A linearly increasing function which turns into a sinusoidal function.
The Laplace transform is
Lcfw_uc (t)f (t
c) =
=
=
Z
Z
Z
1
0
c
1
1
0
=e
cs
=e
cs
e
st
uc (t)f (t
e
st
f (t
c)dt
s(t0 +
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
8.5. SOLUTION OF THE DIFFUSION EQUATION
109
Since v(x) must satisfy the same boundary conditions of u(x, t), we have v(0) =
C1 and v(L) = C2 , and we determine A = C1 and B = (C2 C1 )/L.
We now express u(x, t) as the sum of the known asymptotic stationary
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
64
CHAPTER 5. SERIES SOLUTIONS
Setting coe cients of powers of x to zero, we rst nd a2 = 0, and then obtain
the recursion relation
1
an+3 =
an .
(5.6)
(n + 3)(n + 2)
Three sequences of coe cientsthose starting with either a0 , a1 or a2 decouple.
In partic
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
82
CHAPTER 6. SYSTEMS OF EQUATIONS
Chapter 7
Nonlinear dierential
equations and bifurcation
theory
Reference: Strogatz, Sections 2.2, 2.4, 3.1, 3.2, 3.4, 6.3, 6.4, 8.2
We now turn our attention to nonlinear dierential equations. In particular, we
study ho
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
100
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Dividing by
x ! 0 results in the wave equation
x and taking the limit
utt = c2 uxx ,
where c2 = T /. Since [T ] = ml/t2 and [] = m/l, we have [c2 ] = l2 /t2 so that
c has units of velocity.
8.3
Fourier series
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
46 CHAPTER 3. SECONDORDER ODES, CONSTANT COEFFICIENTS
2
The particular solution given by (3.26), with !0 = k/m, = !/m, f = F/m,
2
2
and tan = !/m(!
!0 ), is the longtime asymptotic solution of the forced,
damped, harmonic oscillator equation (3.24).
We
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
6.2. COUPLED FIRSTORDER EQUATIONS
73
phase space diagram
2
1.5
1
x
2
0.5
0
0.5
1
1.5
2
2
1.5
1
0.5
0
x
0.5
1
1.5
2
1
Figure 6.1: Phase space diagram for example with two real eigenvalues of opposite sign.
Therefore, the equivalent secondorder linear hom
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Chapter 0
A short mathematical
review
A basic understanding of calculus is required to undertake a study of dierential
equations. This zero chapter presents a short review.
0.1
The trigonometric functions
The Pythagorean trigonometric identity is
sin2 x +
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
10
CHAPTER 0. A SHORT MATHEMATICAL REVIEW
Useful trigonometric relations can be derived using ei and properties of the
exponential function. The addition law can be derived from
ei(x+y) = eix eiy .
We have
cos(x + y) + i sin(x + y) = (cos x + i sin x)(cos
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
2.3. LINEAR EQUATIONS
Example: Solve
dy
dx
19
+ 2y = e
x
, with y(0) = 3/4.
Note that this equation is not separable. With p(x) = 2 and g(x) = e
have
Z x
(x) = exp
2dx
x
, we
0
= e2x ,
and
y=e
=e
=e
=e
=e
Example: Solve
dy
dx
Z x
3
2x
x
+
e e dx
4
0
Z x
3
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
3.5. INHOMOGENEOUS ODES
37
Example: Solve x + 2x + x = 0 with x(0) = 1 and x(0) = 0.
The characteristic equation is
r2 + 2r + 1 = (r + 1)2
= 0,
which has a repeated root given by r = 1. Therefore, the general solution to
the ode is
x(t) = c1 e t + c2 te t
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2016
Math2351(2016 Spring): Tutorial note 01
Zhang Lian and
Li Haotian
Review: Introduction of ODE
1. What is ODE?
ODE stands for Ordinary Differential Equations. In other words, equations involve ordinary derivatives.
du
The general form of ODE in this tutori