solve the equation y"+y=0 by power series method.
Answers
Answered by
1
The solution is
y=c0∞∑n=0
x
n
n
!
=
c
0
e
x
,
where
c
0
is any constant.
Let us look at some details.
Let
y
=
∞
∑
n
=
0
c
n
x
n
y
'
=
∞
∑
n
=
1
n
c
n
x
n
−
1
=
∞
∑
n
=
0
(
n
+
1
)
c
n
+
1
x
n
So, we can rewrite
y
'
−
y
=
0
as
∞
∑
n
=
0
(
n
+
1
)
c
n
+
1
x
n
−
∞
∑
n
=
0
c
n
x
n
=
0
by combining the summations,
⇒
∞
∑
n
=
0
[
(
n
+
1
)
c
n
+
1
−
c
n
]
x
n
=
0
so, we have
(
n
+
1
)
c
n
+
1
−
c
n
=
0
⇒
c
n
+
1
=
1
n
+
1
c
n
Let us observe the first few terms.
c
1
=
1
1
c
0
=
1
1
!
c
0
c
2
=
1
2
c
1
=
1
2
⋅
1
1
!
c
0
=
1
2
!
c
0
c
3
=
1
3
c
2
=
1
3
⋅
1
2
!
c
0
=
1
3
!
c
0
.
.
.
c
n
=
1
n
!
c
0
Hence, the solution is
y
=
∞
∑
n
=
0
1
n
!
c
0
x
n
=
c
0
∞
∑
n
=
0
x
n
n
!
=
c
0
e
x
,
where
c
0
is any constant.
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