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
5.4: REGULAR SINGULAR POINTS
KIAM HEONG KWA
Recall that a point x0 is said to be a singular point of the equation
d2 y
dy
(1)
P (x) 2 + Q(x) + R(x)y = 0,
dx
dx
where P (x), Q(x), and R(x) are analytic functions about x0 , provided
one of the functions
Q(x
5.5: EULERS HOMOGENEOUS EQUATIONS
KIAM HEONG KWA
A homogeneous linear equation of the form
dn y
dn1 y
dn2 y
+ 1 xn1 n1 + 2 xn2 n2 + + n y = 0,
dxn
dx
dx
where i R for each i, i = 1, 2, , n, is called an nth order Eulers
equation. By making the substitutio
5.6: SERIES SOLUTIONS NEAR A REGULAR
SINGULAR POINT
KIAM HEONG KWA
In this and the next sections, we indicate the construction of solutions
to the second-order homogeneous linear equation
(1)
P (x)y + Q(x)y + R(x)y = 0
in the vicinity of a regular singula
5.7: SERIES SOLUTIONS NEAR A REGULAR
SINGULAR POINTS (CONTINUED)
KIAM HEONG KWA
Convention 1. We will refer to several equations from section 5.6
very often. When we refer to these equations, we will prepend1 their
labels by the section number. For instan
6.1: THE LAPLACE TRANSFORM
KIAM HEONG KWA
A tool of frequent occurrence in modern mathematics on studying
an unknown function f is to transform the function f into another
function by means of a linear integral transformation T :
b
(1)
f (t) T f (s) = T [
6.2: SOLUTION OF INITIAL VALUE PROBLEMS
KIAM HEONG KWA
In order to solve dierential equations using Laplace transformation,
it is necessary to be able to calculate the Laplace transform Lf of the
derivative f of a function f in terms of Lf . This is the c
6.3: STEP FUNCTION
KIAM HEONG KWA
An important function that occurs in electrical systems is the (delayed) unit step function
(1)
ua (t) =
0 if t < a,
1 if t a,
where a 0. It is also known as Heaviside function. It delays its
output until t = a and then a
DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS
FORCING FUNCTIONS
KIAM HEONG KWA
Consider the initial value problem
(1)
y (t) + y (t) + y (t) = ua (t), y (0) = y0 , y (0) = y0 ,
where , R and ua is the Heaviside function
(2)
ua (t) =
0 if t < a,
1 if t a
for so
6.5: IMPLUSE FUNCTIONS
KIAM HEONG KWA
For a given t-value, say t0 , consider the family of functions
1
2
(t t0 ) =
(1)
0
if t0 < t < t0 + ,
if t t0 or t t0 + ,
that are indexed by > 0. These functions are known as impulse
functions centered at t0 . Physi
6.6: THE CONVOLUTION INTEGRAL
KIAM HEONG KWA
Let f and g be two functions on [0, ). The convolution of these
functions is the function
t
(1)
f ( )g (t ) d
(f g )(t) =
0
whenever the integral is dened. It is easy to see that the convolution
is commutative
Homework List for Math 255.01 (Winter 2011)
Kiam Heong Kwa
February 28, 2011
Remark 1. This list will be updated throughout the quarter.
Remark 2. A problem with an asterisk (*) is relatively more difcult.
Remark 3. Turn in only boxed problems.
Final is o
POWER SERIES SOLUTIONS OF SECOND-ORDER
LINEAR EQUATIONS
KIAM HEONG KWA
1. Method of Undetermined Coefficients
Consider a general second-order linear homogeneous equation
(1.1)
y + p(x)y + q (x)y = 0,
where the coecients are real analytic in the sense that
5.2: SERIES SOLUTIONS NEAR AN ORDINARY
POINT
KIAM HEONG KWA
Consider the dierential equation
d2 y
dy
(1)
P (x) 2 + Q(x) + R(x)y = 0,
dx
dx
where P (x), Q(x), and R(x) are general analytic functions1. A point
x0 at which both Q(x)/P (x) and R(x)/P (x) are
5.1: REVIEW OF POWER SERIES
KIAM HEONG KWA
For a sequence of (real) numbers cfw_an and a xed (real) number
n=0
x0 , a series of the form
an (x x0 )n
(1)
n=0
is called a power series of the variable x centered at x0 . It denes a
function f (x) of x whenev
4.4: THE METHOD OF VARIATION OF PARAMETERS
KIAM HEONG KWA
The goal of this section is to generalize the method of variation of
parameters for second-order linear equations in section 3.7 to the class
of higher order linear equations. For concreteness, we
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
2.7: NUMERICAL APPROXIMATIONS:
EULERS METHOD
KIAM HEONG KWA
Unless otherwise stated, we shall assume that the rst-order initial
value problem
dy
= f (t, y ), y (t0 ) = y0 ,
(1)
dt
has a unique solution (t) in an interval containing t0 . We have indicated
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
3.6: NONHOMOGENEOUS EQUATIONS AND
METHOD OF UNDETERMINED COEFFICIENTS
KIAM HEONG KWA
We would like to construct the general solution of the nonhomogeneous equation
(1)
ay + by + cy = g (t),
where a, b, c R with a = 0 and g (t) is a smooth function, i.e.,
3.7: VARIATION OF PARAMETERS
KIAM HEONG KWA
This section exhibits the method of variation of parameters to compute the general solution of a second-order linear nonhomogeneous
equation from a given fundamental set of solutions of its associated
homogeneou
4.1: GENERAL THEORY OF HIGHER ORDER LINEAR
EQUATIONS
KIAM HEONG KWA
An nth order linear dierential equation is an equation of the form
dn y
dn1 y
dy
+ P1 (t) n1 + + Pn1 (t) + Pn (t)y = G(t),
n
dt
dt
dt
where Pi (t), i = 0, 1, , n, and G(t) are continuous