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Review Problems
its degree, unless it’s the zero polynomial, we conclude that the linear combination (6.27)
vanishes identically on (
−∞
,
∞
) if and only if
c
1
=
c
2
=
c
3
=
c
4
= 0. This means that the
given functions are linearly independent on (
−∞
,
∞
).
5. (a)
Solving the auxiliary equation yields
(
r
+5)
2
(
r
−
2)
3
(
r
2
+1)
2
=0
⇒
(
r
2
=0 or
(
r
−
2)
3
(
r
2
2
.
Thus, the roots of the auxiliary equation are
r
=
−
5
of multiplicity 2
,
r
= 2
of multiplicity 3
,
r
=
±
i
of multiplicity 2
.
According to (22) on page 329 and (28) on page 330 of the text, the set of functions
(assuming that
x
is the independent variable)
e
−
5
x
,x
e
−
5
x
,e
2
x
e
2
x
2
e
2
x
,
cos
x, x
cos
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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