Chapter 6
Equation (6.5) will equal zero and, therefore, the differential equation will be satisfied for
r
=
±
1 and
r
= 2. Thus, three solutions to the differential equation are
y
=
x
,
y
=
x
−
1
,
and
y
=
x
2
.
Since these functions are linearly independent, they form a fundamental
solution set.
(b)
Let
y
(
x
) =
x
r
. In addition to (6.4), we need the fourth derivative of
y
(
x
).
y
(4)
= (
y
) =
r
(
r
−
1)(
r
−
2)(
r
−
3)
x
r
−
4
= (
r
4
−
6
r
3
+ 11
r
2
−
6
r
)
x
r
−
4
.
Thus, if
y
=
x
r
is a solution to this fourth order CauchyEuler equation, then we must
have
x
4
(
r
4
−
6
r
3
+ 11
r
2
−
6
r
)
x
r
−
4
+ 6
x
3
(
r
3
−
3
r
2
+ 2
r
)
x
r
−
3
+2
x
2
(
r
2
−
r
)
x
r
−
2
−
4
xrx
r
−
1
+ 4
x
r
= 0
⇒
(
r
4
−
6
r
3
+ 11
r
2
−
6
r
)
x
r
+ 6(
r
3
−
3
r
2
+ 2
r
)
x
r
+ 2(
r
2
−
r
)
x
r
−
4
rx
r
+ 4
x
r
= 0
⇒
(
r
4
−
5
r
2
+ 4)
x
r
= 0
.
(6.6)
Therefore, in order for
y
=
x
r
to be a solution to the equation with
x >
0, we must have
r
4
−
5
r
2
+ 4 = 0. Factoring this equation yields
r
4
−
5
r
2
+ 4 = (
r
2
−
4)(
r
2
−
1) = (
r
−
2)(
r
+ 2)(
r
−
1)(
r
+ 1) = 0
.
Equation (6.6) will be satisfied if
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Derivative, XR

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