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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 2013
Ch 2.2: Separable Equations
In this section we examine a subclass of linear and nonlinear
first order equations. Consider the first order equation
dy
= f ( x, y )
dx
We can rewrite this in the form
dy
M ( x, y ) + N ( x, y ) = 0
dx
For example, let M(x,y)
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 1.1:
Basic Mathematical Models; Direction Fields
Differential equations are equations containing derivatives.
Derivatives describe rates of change.
The following are examples of physical phenomena involving
rates of change:
Motion of fluids
Motion of m
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 3.4:
Repeated Roots; Reduction of Order
Recall our 2nd order linear homogeneous ODE
ay + by + cy = 0
where a, b and c are constants.
Assuming an exponential soln leads to characteristic equation:
y(t ) = ert ar 2 + br + c = 0
Quadratic formula (or fact
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 3.5: Nonhomogeneous Equations;
Method of Undetermined Coefficients
Recall the nonhomogeneous equation
y + p(t ) y + q(t ) y = g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y + p(t ) y + q(t
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 1.2:
Solutions of Some Differential Equations
Recall the free fall and owl/mice differential equations:
v = 9.8  0.2v,
p = 0.5 p  450
These equations have the general form y' = ay  b
We can use methods of calculus to solve differential
equations of
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 3.1: Second Order Linear Homogeneous
Equations with Constant Coefficients
A second order ordinary differential equation has the
general form
y = f (t , y, y)
where f is some given function.
This equation is said to be linear if f is linear in y and y':
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 2.1: Linear Equations;
Method of Integrating Factors
A linear first order ODE has the general form
dy
= f (t , y )
dt
where f is linear in y. Examples include equations with
constant coefficients, such as those in Chapter 1,
y = ay + b
or equations wi
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Ch 3.2: Solutions of Linear Homogeneous Equations; Wronskian
Let p, q be continuous functions on an interval I = (a, b),
which could be infinite. For any function y that is twice
differentiable on I, define the differential operator L by
L[y] = y + p y +
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
HKUST
MATH150 Introduction to Differential Equations
Final Examination (Version White)
Name:
15th December 2004
8:3010:30
Student I.D.:
Tutorial Section:
Directions:
Write your name, ID number, and tutorial section in the space provided above.
DO NOT op
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math150L2 Final Exam Answers, Spring 06
Part I: MC Questions
Version A
Question
1
2
3
4
5
6
7
8
9
Answer
d
b
d
c
e
a
e
b
d
Question
1
2
3
4
5
6
7
8
9
Answer
c
e
c
a
b
d
b
e
c
Total
Version B
Total
Answers of Version A MC Questions
1. Suppose that the pop
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH150 Introduction to Ordinary Differential Equations
Suggested Solution to Final 2005
1.
(a) Solve the initial value problem for x 0 :
dy 2 cos 2 x
=
, y (0) = 2 .
dx
3+ 2y
(b) Determine where the solution attains its minimum value. What is the value o
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
HKUST
MATH150 Introduction to Differential Equations
Final Examination (Version A)
Name:
20th May 2005
16:3018:30
Student I.D.:
Tutorial Section:
Directions:
Write your name, ID number, and tutorial section in the space provided above.
DO NOT open the e
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math150 Introduction to Ordinary Differential Equations, Spring 05
Final Examination Solution: Version A
Part I: Multiple Choice Question.
Question
1
2
3
4
5
6
7
8
Answer
d
c
b
a
c
d
c
c
Total
1. Which of the following functions is an integrating factor f
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 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 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 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 2013
Ch 3.3:
Complex Roots of Characteristic Equation
Recall our discussion of the equation
ay + by + cy = 0
where a, b and c are constants.
Assuming an exponential soln leads to characteristic equation:
y(t ) = ert ar 2 + br + c = 0
Quadratic formula (or fact
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Assignment 5
Exercise 7.5: 7, 9
Exercise 7.6: 9,10
Exercise 7.5
(a) Find the general solution of the given system of equations.
(b) Draw a direction field and a few of the trajectories. The coefficient matrix has a
zero eigenvalue. As a result, the
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 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
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
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
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
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
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
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
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
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