1.1: SOME BASIC MATHEMATICAL MODELS
KIAM HEONG KWA
A dierential equation is an equation between specied derivatives of
an unknown function, its values, and known quantities and functions.
The equation is called
an ordinary dierential equation (ODE) if th
10.1: TWO-POINT BOUNDARY VALUE PROBLEMS
KIAM HEONG KWA
1. Two-Point Boundary Value Problems
Roughly speaking, a two-point boundary value problem consists of a
dierential equation together with suitable boundary conditions. The
boundary conditions can eith
10.2: FOURIER SERIES
KIAM HEONG KWA
Historical note. On December 21, 1807, an engineer named Joseph Fourier
announced to the prestigious French Academy of Sciences that an arbitrary
function could be expanded in an innite series of sines and cosines1. His
10.4: EVEN AND ODD FUNCTIONS
KIAM HEONG KWA
1. The Fourier Cosine and Sine Series
It is often desired to expand in a Fourier series of period 2L a function f (x) originally dened only on either one of the intervals (0, L),
(0, L], [0, L), and [0, L]. One
10.5: SEPARATION OF VARIABLES AND HEAT
CONDUCTION IN A ROD
KIAM HEONG KWA
The variation of temperature u(x, t) in a bar whose axis lies along
the x-axis is governed by the heat conduction equation
(1)
2 uxx = ut , 0 < x < L, t > 0,
where the thermal diusi
10.6: OTHER HEAT CONDUCTION PROBLEMS
KIAM HEONG KWA
1. Non-homogeneous Boundary Conditions
Recall that the solution to the heat conduction problem
(1.1a)
2 uxx = ut , 0 < x < L, t > 0,
with the initial condition
(1.1b)
u(x, 0) = f (x), 0 x L,
where f (x)
10.7: THE WAVE EQUATION: VIBRATIONS OF AN
ELASTIC STRING
KIAM HEONG KWA
Consider an elastic string of length L which is tightly stretched between two supports at the same horizontal level, so that it lies along
the x-axis. Set the string in motion such th
11/7/2010
Answers to Problems
Chapter 10
Section 10.1, page 575.
1.
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3.
for all L;
if
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if
; no solution if
5.
N o solution
6.
7.
N o solution
8.
9.
10.
11.
12.
13.
N o solution
14.
15.
16.
17.
18.
19.
20.
21.
(a)
wiley.com/Ym95Y2IwMS54Zm9ybQ.e
103/122
CANONICAL FORMS OF 2 2 MATRICES
AND THEIR APPLICATIONS
KIAM HEONG KWA
1. Canonical Forms of 2 2 Matrices
The characteristic equation of a 2 2 complex matrix, say
A=
a11 a12
,
a21 a22
is the quadratic equation
det(A I2 ) = 2 (tr A) + det A = 0,
where I2 =
AN INSTANCE OF GIBBS PHENOMENON
KIAM HEONG KWA
The Gibbs phenomenon, named after the American physicist Josiah
Willard Gibbs, is the peculiar manner in which the Fourier series of
a piecewise continuously dierentiable periodic function behaves at a
jump d
Homework List for Math 415.01/415.02 (Spring
2010)
Kiam Heong Kwa
May 27, 2010
Remark 1. This list will be updated throughout the quarter.
Remark 2. A problem with an asterisk (*) is relatively more difcult.
Sec 7.8 2, 4, 8, 10, 15*
Problems 2, 4, 8, 10:
SEPARATION AND COMPARISON THEOREMS
KIAM HEONG KWA
Recall that a zero of a function is a point where its value is zero. As
an application of the Wronskian, we use it to separate and compare
zeros of solutions to second-order linear homogeneous equations.
T
7.5/7.6: HOMOGENEOUS LINEAR SYSTEMS WITH
CONSTANT COEFFICIENTS
KIAM HEONG KWA
A matrix is said to be complex if all its entries are complex scalars.
Likewise, a matrix is said to be real if all its entries are real scalars.
Except row vectors and column v
7.4: BASIC THEORY OF SYSTEMS OF FIRST ORDER
LINEAR EQUATIONS
KIAM HEONG KWA
1. The Wronskian of Solutions and The Liouvilles
Formula
Consider the homogeneous system of n rst order linear equations
(1.1)
x = P (t)x,
where
x1
p11 (t) p12 (t)
x2
p21 (t) p
2.1: FIRST-ORDER LINEAR EQUATIONS
KIAM HEONG KWA
An nth order ODE is called linear if one can write it in the form
(1)
a0 (t)y (n) + a1 (t)y (n1) + + an1 (t)y + an (t)y = g (t),
where the coecients ai (t) (i = 0, 1, , n) and g (t) are known functions; oth
2.2: SEPARABLE EQUATIONS
KIAM HEONG KWA
A separated equation is a rst-order equation of the form
dy
= 0,
dx
where M (x) and N (y ) are continuous functions on some (open) intervals. It is understood that x is the independent variable and y is the
unknown
2.4: DIFFERENCES BETWEEN LINEAR
AND NONLINEAR EQUATIONS
KIAM HEONG KWA
We have so far considered a number of initial-value problems of the
form
(1)
y = f (t, y ), y (t0 ) = y0 ,
where the rate function f (t, y ) is continuously dierentiable (within its
do
2.5: AUTONOMOUS EQUATIONS
AND POPULATION DYNAMICS
KIAM HEONG KWA
We will dene the (asymptotic) stability of an equilibrium solution
of an autonomous rst order equation. We will also state two stability
tests for testing the (asymptotic) stability of such
2.6: EXACT EQUATIONS
KIAM HEONG KWA
1. Exact Equations and Integrating Factors
A rst-order equation is an equation that can be written in the form
(1.1)
M (x, y ) + N (x, y )
dy
= 0,
dx
where M (x, y ) and N (x, y ) are continuous functions of two real va
3.1/3.4/3.5: HOMOGENEOUS EQUATIONS
WITH CONSTANT COEFFICIENTS
KIAM HEONG KWA
A general second-order dierential equation has the form
d2 y
=f
dt2
(1)
t, y,
dy
dt
,
where f is a given function of three variables. Second-order equations
arise very often in a
3.2: FUNDAMENTAL SOLUTIONS OF LINEAR
HOMOGENEOUS EQUATIONS
KIAM HEONG KWA
We have learned in the last section that there are always two linearly
independent solutions y1 (t) and y2 (t) to a second-order homogeneous
linear equation
(1)
ay + by + cy = 0
wit
3.3: LINEAR INDEPENDENCE
AND THE WRONSKIAN
KIAM HEONG KWA
Generally, two functions f (t) and g (t) are said to be linearly independent on an interval I if the linear relation
(1)
c1 f (t) + c2 g (t) = 0,
where c1 and c2 are constant scalars, holds for all
3.5: REDUCTION OF ORDER
KIAM HEONG KWA
Consider a general second-order linear homogeneouos equation
(1)
y + p(t)y + q (t)y = 0
with continuous coecients on an open interval I. Provided with a
nonzero solution y1 (t), we want to calculate the general solut
3.8: MECHANICAL AND ELECTRICAL VIBRATIONS
KIAM HEONG KWA
1. Mechanical Vibrations
Consider a mass m attached to an elastic spring1 of length l, which
is suspended from a rigid horizontal support. The equilibrium position
of the mass is the point where the
3.9: FORCED VIBRATIONS
KIAM HEONG KWA
1. Forced Vibrations with Damping
Continuing our investigation of the spring-mass system with damping, we now suppose that a periodic external force F (t) = F0 cos t is
applied to the mass. Here F0 and are numerically
7.1: AN INTRODUCTION TO SYSTEMS OF FIRST
ORDER LINEAR EQUATIONS
KIAM HEONG KWA
The Notation
For notational convenience, we denote vectors by boldfaced lowercase Latin letters x, y , etc. or lower-case Greek letters , , etc. We
also identify vectors to col