MAT 127 PRACTICE FINAL
(1)
Consider the initial value problem
y
00

y
0
+ 3
y
= 0
y
(0) = 1
, y
0
(0) =

1
.
Assuming the solution to this initial value problem has is the power series
y
=
∞
X
n
=0
c
n
x
n
,
find all the coeffiecients
c
n
for
n
≤
6.
Solution:
We have that
y
00
=
∞
X
n
=0
(
n
+ 2)(
n
+ 1)
c
n
+2
x
n
,

y
0
=
∞
X
n
=0

(
n
+ 1)
c
n
+1
x
n
,
3
y
=
∞
X
n
=0
3
c
n
x
n
.
If we add these series together (adding term by term) we must get the power
series
∑
∞
n
=0
0
x
n
(this is what the differential equation states). Thus, for all
n
≥
0 we have that
(1)
(
n
+ 2)(
n
+ 1)
c
n
+2

(
n
+ 1)
c
n
+1
+ 3
c
n
= 0
.
On the other hand, we deduce from
y
(0) = 1 and from
y
0
(0) =

1 that
(2)
c
0
= 1
and
(3)
c
1
=

1
.
To compute
c
2
we use properties (1)(3) (with
n
= 0 in (1)) to get
2
c
2

(

1) + 3 = 0
from which we can solve for
(4)
c
2
=

2
.
To compute
c
3
we use properties (1),(3),(4) (with
n
= 1 in (1)) to get
6
c
3

2(

2) + 3(

1) = 0
1
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MAT 127 PRACTICE FINAL
from which we can solve for
(5)
c
3
=

1
/
6
.
To compute
c
4
we use properties (1)(4)(5) (with
n
= 2 in (1))
.....
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 Fall '07
 GuanYuShi
 Power Series, Mathematical Series, 2 hours, 3 hours

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