23
Variation of Parameters
(A Better Reduction of Order Method for
Nonhomogeneous Equations)
Variation of parameters is another way to solve nonhomogeneous linear differential equations,
be they second order,
ay + by + cy = g ,
or even higher order,
a0 y
28
Piecewise-Dened Functions and
Periodic Functions
At the start of our study of the Laplace transform, it was claimed that the Laplace transform is
particularly useful when dealing with nonhomogeneous equations in which the forcing functions are not cont
32
Power Series Solutions II:
Generalizations and Theory
A major goal in this chapter is to conrm the claims made in theorems 31.1 and 31.3 regarding the
validity of the algebraic method. Along the way, we will also expand both the set of differential
equ
31
Power Series Solutions I: Basic
Computational Methods
When a solution to a differential equation is analytic at a point, then that solution can be represented by a power series about that point. In this and the next chapter, we will discuss when
this c
Authors Foreword
This textbook reects my personal views on how an introductory course in ordinary differential
equations really should be taught, tempered by the constraints of time, the level and interests of
the students, and other practical concerns. I
Answers to Selected Exercises
Chapter 30
2a.
121
81
2b.
2h. 14
4c.
n =4
9,841
6,561
2i. 5
n =1
n =0
5b.
6c.
k =0
k =0
1
(k
2
4a. 1 +
4d.
n =1
n =3
n =2
4g.
n =4
k
6d.
9d.
k =0
k+1 k
x
2k +1
with R =
8/14/2011
32
x
8
k =1
2
4b. x +
4e. 6 +
n =1
k =4
( 1)k
13
Reduction of Order
We shall take a brief break from developing the general theory for linear differential equations
to discuss one method (the reduction of order method) for nding the general solution to any
linear differential equation. In some ways,
4
Separable First-Order Equations
As we will see below, the notion of a differential equation being separable is a natural generalization of the notion of a rst-order differential equation being directly integrable. Whats more,
a fairly natural modication
1
A Guide to Using This Text
What follows is a suggested schedule of the chapters and sections to be covered (and not covered)
in a typical introductory course on differential equations, along with some commentary on the
material. This discussion is direc
ErratA
IN
Ordinary Differential Equations:
An Introduction to the Fundamentals
(June 12, 2011)
Notes:
text from the book is printed using this font (Times Roman)
Comments are printed using this font (Univers Condensed)
text to be added is underlined and i
6
Simplifying Through Substitution
In previous chapters, we saw how certain types of rst-order differential equations (directly
integrable, separable, and linear equations) can be identied and put into forms that can be
integrated with relative ease. In t
1
The Starting Point:
Basic Concepts and Terminology
Let us begin our study of differential equations with a few basic questions questions that
any beginner should ask:
What are differential equations?
What can we do with them? Solve them? If so, what do
22
Springs: Part II (Forced Vibrations)
Let us look, again, at those mass/spring systems discussed in chapter 17. Remember, in such a
system we have a spring with one end attached to an immobile wall and the other end attached to
some object that can move
17
Springs: Part I
Second-order differential equations arise in a number of applications. We saw one involving a
falling object at the beginning of this text (the falling frozen duck example in section 1.2). In
fact, since acceleration is given by the sec
16
Second-Order Homogeneous Linear
Equations with Constant Coefcients
A very important class of second-order homogeneous linear equations consists of those with
constant coefcients; that is, those that can be written as
ay + by + cy = 0
where a , b and c
8
Slope Fields: Graphing Solutions
Without the Solutions
Up to now, our efforts have been directed mainly towards nding formulas or equations describing
solutions to given differential equations. Then, sometimes, we sketched the graphs of these
solutions
30
Series Solutions: Preliminaries
(A Brief Review of Innite Series, Power
Series and a Little Complex Variables)
At this point, you should have no problem in solving any differential equation of the form
a
d2 y
dy
+b
+ cy = 0
dx
dx 2
ax 2
or
d2 y
dy
+ bx
20
Nonhomogeneous Equations in General
Now that we know how to solve a couple of rather broad classes of homogeneous equations, it
is time to start looking at nonhomogeneous equations.
20.1
Basic Theory
Recollections about Linearity
Let us go back to our
10
The Art and Science of Modeling with
First-Order Equations
For some, modeling is the building of small plastic replicas of famous ships; for others,
modeling means standing in front of cameras wearing silly clothing; for us, modeling is
the process of
12
Higher-Order Linear Equations:
Introduction and Basic Theory
We have just seen that some higher-order differential equations can be solved using methods
for rst-order equations after applying the substitution v = dy/dx . Unfortunately, this approach
ha
33
Modied Power Series Solutions and the
Basic Method of Frobenius
The partial sums of a power series solution about an ordinary point x0 of a differential equation
provide fairly accurate approximations to the equations solutions at any point x near x0 .
11
Higher-Order Equations: Extending
First-Order Concepts
Let us switch our attention from rst-order differential equations to differential equations of
order two or higher. Our main interest will be with second-order differential equations, both
because
7
The Exact Form and General
Integrating Factors
In the previous chapters, weve seen how separable and linear differential equations can be solved
using methods for converting them to forms that can be easily integrated. In this chapter, we will
develop a
9
Eulers Numerical Method
In the last chapter, we saw that a computer can easily generate a slope eld for a given rst-order
differential equation. Using that slope eld we can sketch a fair approximation to the graph
of the solution y to a given initial-va