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Exercises 5.2
If ∆
≥
0, then
√
∆
<

λ
+1

and a fundamental solution set is
{
e
r
1
t
,e
r
2
t
}
,
if ∆
>
0
,
{
e
r
1
t
,te
r
1
t
}
,
if ∆ = 0
,
(5.16)
where both roots
r
1
,
r
2
are nonpositive if and only if
λ
≤−
1. For
λ
=
−
1w
ehav
e
∆=(
−
1+1)
2
−
4(
−
1+3)
<
0, and we have a particular case of the fundamental
solution set (5.15) (without exponential term) consisting of bounded functions. Finally,
if
λ<
−
1, then
r
1
<
0,
r
2
≤
0, and all the functions listed in (5.15), (5.16) are bounded.
Any solution
x
(
t
) is a linear combination of fundamental solutions and, therefore, all solutions
x
(
t
) are bounded if and only if
−
3
≤
λ
≤−
1.
31.
Solving this problem, we follow the arguments described in Section 5.1, page 242 of the text,
i.e.,
x
(
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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