POWER SERIES
Solving a second order homogeneous differential equation using the power series
17.
p479
′′ +
′ 
=
y
xy
y
2
0
y
y
( )
,
( )
0
1
0
0
=
′
=
To solve this problem we will:
1. Set
y
equal to a power series and substitute the power series and its
derivatives into the equation for
y
and the derivatives of
y
.
2. Manipulate the expression so that
x
and the summation operators
can be eliminated.
3. Form a
recurrence relation
from this expression by solving for a
constant other than
c
n
.
4. Substitute values for
n
into the
recurrence relation
to form two
new series expressions involving two independent constants.
This
is the general solution in rough form.
5. Simplify these series as sums where possible by recognizing
patterns which can be expressed as sums.
6. Substitute the first initial value into the solution and substitute the
second initial value into its derivative to determine the values of
the two constants.
7. Substitute the values of the two constants into the general solution
to find the particular solution.
let
y
c x
n
n
n
=
=
∞
∑
0
then
′ =

=
∞
∑
y
nc x
n
n
n
1
1
and
′′ =


=
∞
∑
y
n n
c x
n
n
n
(
)
1
2
2
Substitute these expressions into the original
equation
n n
c x
x
nc x
c x
n
n
n
n
n
n
n
n
n
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 Spring '08
 comech
 Addition, Derivative, Power Series, Summation, NC, general solution

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