James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 17.1 Second-Order Linear Equations 1157
In the first term, 2ar 1 b − 0 by Equations 9; in the second term, ar 2 1 br 1 c − 0
because r is a root of the auxiliary equation. Since y 1 − e rx and y 2 − xe rx are linearly
independent solutions, Theorem 4 provides us with the general solution.
10 If the auxiliary equation ar 2 1 br 1 c − 0 has only one real root r, then
the general solution of ay0 1 by9 1 cy − 0 is
y − c 1 e rx 1 c 2 xe rx
Figure 2 shows the basic solutions
f sxd − e 23xy2 and tsxd − xe 23xy2 in
Exam ple 3 and some other members of
the family of solutions. Notice that all
of them approach 0 as x l `.
f
f-g
8
5f+g
f+5g
_2 2
f+g
g-f
g
Example 3 Solve the equation 4y0 1 12y9 1 9y − 0.
SOLUtion The auxiliary equation 4r 2 1 12r 1 9 − 0 can be factored as
s2r 1 3d 2 − 0
so the only root is r − 2 3 2 . By (10) the general solution is
y − c 1 e 23xy2 1 c 2 xe 23xy2
Case iii b 2 2 4ac , 0
In this case the roots r 1 and r 2 of the auxiliary equation are complex numbers. (See
Appen dix H for information about complex numbers.) We can write
■
FIGURE 2
_5
r 1 − 1 i
r 2 − 2 i
where and are real numbers. [In fact, − 2bys2ad, − s4ac 2 b 2 ys2ad.] Then,
using Euler’s equation
e i − cos 1 i sin
from Appendix H, we write the solution of the differential equation as
y − C 1 e r 1 x 1 C 2 e r 2 x − C 1 e s1idx 1 C 2 e s2idx
− C 1 e x scos x 1 i sin xd 1 C 2 e x scos x 2 i sin xd
− e x fsC 1 1 C 2 d cos x 1 isC 1 2 C 2 d sin xg
− e x sc 1 cos x 1 c 2 sin xd
where c 1 − C 1 1 C 2 , c 2 − isC 1 2 C 2 d. This gives all solutions (real or complex) of the
dif ferential equation. The solutions are real when the constants c 1 and c 2 are real. We
summarize the discussion as follows.
11 If the roots of the auxiliary equation ar 2 1 br 1 c − 0 are the complex
numbers r 1 − 1 i, r 2 − 2 i, then the general solution of
ay0 1 by9 1 cy − 0 is
y − e x sc 1 cos x 1 c 2 sin xd
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.