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CHAPTER 2
SecondOrder Linear ODEs
Major Changes
Among linear ODEs those of second order are by far the most important ones from the
viewpoint of applications, and from a theoretical standpoint they illustrate the theory of
linear ODEs of any order (except for the role of the Wronskian). For these reasons we
consider linear ODEs of third and higher order in a separate chapter, Chap. 3.
The new Sec. 2.2 combines all three cases of the roots of the characteristic equation of
a homogeneous linear ODE with constant coefficients. (In the last edition the complex
case was discussed in a separate section.)
Modeling applications of the method of undetermined coefficients (Sec. 2.7) follow
immediately after the derivation of the method (mass–spring systems in Sec. 2.8, electric
circuits in Sec. 2.9), before the discussion of variation of parameters (Sec. 2.10).
The new Sec. 2.9 combines the old Sec. 1.7 on modeling electric circuits by firstorder
ODEs and the old Sec. 2.12 on electric circuits modeled by secondorder ODEs. This
avoids discussing the physical aspects and foundations twice.
SECTION 2.1. Homogeneous Linear ODEs of SecondOrder, page 45
Purpose.
To extend the basic concepts from firstorder to secondorder ODEs and to
present the basic properties of linear ODEs.
Comment on the Standard Form (1)
The form (1), with 1 as the coefficient of
y
0
, is practical, because if one starts from
ƒ(
x
)
y
0 1
g
(
x
)
y
9 1
h
(
x
)
y
5
r

(
x
),
one usually considers the equation in an interval
I
in which ƒ(
x
) is nowhere zero, so that
in
I
one can divide by ƒ(
x
) and obtain an equation of the form (1). Points at which
ƒ(
x
)
5
0 require a special study, which we present in Chap. 5.
Main Content, Important Concepts
Linear and nonlinear ODEs
Homogeneous linear ODEs (to be discussed in Secs. 2.1
2
2.6)
Superposition principle for homogeneous ODEs
General solution, basis, linear independence
Initial value problem (2), (4), particular solution
Reduction to first order (text and Probs. 15
2
22)
Comment on the Three ODEs after (2)
These are for illustration, not for solution, but should a student ask, answers are that the
first will be solved by methods in Sec. 2.7 and 2.10, the second is a Bessel equation
(Sec. 5.5) and the third has the solutions
6
Ï
c
1
x
1
w
c
2
w
with any
c
1
and
c
2
.
Comment on Footnote 1
In 1760, Lagrange gave the first methodical treatment of the calculus of variations. The
book mentioned in the footnote includes all major contributions of others in the field and
made him the founder of analytical mechanics.
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Page 30
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View Full DocumentComment on Terminology
p
and
q
are called the
coefficients
of (1) and (2). The function
r
on the right is
not
called
a coefficient, to avoid the misunderstanding that
r
must be constant when we talk about
an ODE
with constant coefficients
.
SOLUTIONS TO PROBLEM SET 2.1, page 52
2.
cos 5
x
and sin 5
x
are linearly independent on any interval because their quotient,
cot 5
x
, is not constant. General solution:
y
5
a
cos 5
x
1
b
sin 5
x
.
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 Spring '08
 KABA

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