CHAPTER 3
Higher Order Linear ODEs
This chapter is new. Its material is a rearranged and somewhat extended version of material
previously contained in some of the sections of Chap 2. The rearrangement is such that
the presentation parallels that in Chap. 2 for secondorder ODEs, to facilitate comparisons.
Root Finding
For higher order ODEs you may need Newton’s method or some other method from
Sec. 19.2 (which is independent of other sections in numerics) in work on a calculator
or with your CAS (which may give you a rootfinding method directly).
Linear Algebra
The typical student may have taken an elementary linear algebra course simultaneously
with a course on calculus and will know much more than is needed in Chaps. 2 and 3.
Thus Chaps. 7 and 8 need not be taken before Chap. 3.
In particular, although the Wronskian becomes useful in Chap. 3 (whereas for
n
5
2
one hardly needs it), a very modest knowledge of determinants will suffice. (For
n
5
2
and 3, determinants are treated in a reference section, Sec. 7.6.)
SECTION 3.1. Homogeneous Linear ODEs, page 105
Purpose.
Extension of the basic concepts and theory in Secs. 2.1 and 2.6 to homogeneous
linear ODEs of any order
n.
This shows that practically all the essential facts carry over
without change. Linear independence, now more involved as for
n
5
2, causes the
Wronskian to become indispensable (whereas for
n
5
2 it played a marginal role).
Main Content, Important Concepts
Superposition principle for the homogeneous ODE (2)
General solution, basis, particular solution
General solution of (2) with continuous coefficients exists.
Existence and uniqueness of solution of initial value problem (2), (5)
Linear independence of solutions, Wronskian
General solution includes all solutions of (2).
Comment on Order of Material
In Chap. 2 we first gained practical experience and skill and presented the theory of the
homogeneous linear ODE at the end of the discussion, in Sec. 2.6. In this chapter, with
all the experience gained on secondorder ODEs, it is more logical to present the whole
theory at the beginning and the solution methods (for linear ODEs with constant
coefficients) afterward. Similarly, the same logic applies to the nonhomogeneous linear
ODE, for which Sec. 3.3 contains the theory as well as the solution methods.
SOLUTIONS TO PROBLEM SET 3.1, page 111
2.
Problems 1–5 should give the student a first impression of the changes occurring in
the transition from
n
5
2 to general
n.
59
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Page 59
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Let
y
1
5
x
1
1,
y
2
5
x
1
2,
y
3
5
x.
Then
y
2
2
2
y
1
1
y
3
5
0
shows linear dependence.
10.
Linearly independent
12.
Linear dependence, since one of the functions is the zero function
14.
cos 2
x
5
cos
2
x
2
sin
2
x
; linearly dependent
16.
(
x
2
1)
2
2
(
x
1
1)
2
1
4
x
5
0; linearly dependent
18.
Linearly independent
20. Team Project. (a)
(1) No. If
y
1
;
0, then (4) holds with any
k
1
Þ
0 and the other
k
j
all zero.
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 Spring '08
 KABA
 Linear Algebra, Linear Independence, Vector Space, general solution, characteristic equation

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