Using Series to Solve Differential Equations
Many differential equations can’t be solved explicitly in terms of finite combinations of
simple familiar functions. This is true even for a simple-looking equation like
y 2xy y 0
1
But it is important to be able to solve equations such as Equation 1 because they arise from
physical problems and, in particular, in connection with the Schrödinger equation in quantum mechanics. In such a case we use the method of power series; that is, we look for a
solution of the form
y f x cx
n
n
c0 c1 x c2 x 2 c3 x 3 n0
The method is to substitute this expression into the differential equation and determine the
values of the coefficients c0 , c1, c2 , . . . .
Before using power series to solve Equation 1, we illustrate the method on the simpler
equation y y 0 in Example 1.
EXAMPLE 1 Use power series to solve the equation y y 0.
SOLUTION We assume there is a solution of the form
2
y c0 c1 x c2 x 2 c3 x 3 cx
n
n
n0
We can differentiate power series term by term, so
nc x
y c1 2c2 x 3c3 x 2 n
n1
n1
3
y 2c2 2 3c3 x nn 1c x
n
n2
n2
In order to compare the expressions for y and y more easily, we rewrite y as follows:
■ ■ By writing out the first few terms of (4), you
can see that it is the same as (3). To obtain (4)
we replaced n by n 2 and began the summation at 0 instead of 2.
y 4
n 2n 1c
n2
xn
n0
Substituting the expressions in Equations 2 and 4 into the differential equation, we
obtain
n 2n 1c
xn n2
n0
cx
n
n
0
n0
or
5
n 2n 1c
n2
cn x n 0
n0
If two power series are equal, then the corresponding coefficients must be equal. Therefore, the coefficients of x n in Equation 5 must be 0:
n 2n 1cn2 cn 0
1
2 ■ USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS
cn2 6
cn
n 1n 2
n 0, 1, 2, 3, . . .
Equation 6 is called a recursion relation. If c0 and c1 are known, this equation allows
us to determine the remaining coefficients recursively by putting n 0, 1, 2, 3, . . . in
succession.
Put n 0:
c2 c0
12
Put n 1:
c3 c1
23
Put n 2:
c4 c2
c0
c0
34
1234
4!
Put n 3:
c5 c3
c1
c1
45
2345
5!
Put n 4:
c6 c4
c0
c0
56
4! 5 6
6!
Put n 5:
c7 c5
c1
c1
67
5! 6 7
7!
By now we see the pattern:
For the even coefficients, c2n 1n
c0
2n!
For the odd coefficients, c2n1 1n
c1
2n 1!
Putting these values back into Equation 2, we write the solution as
y c0 c1 x c2 x 2 c3 x 3 c4 x 4 c5 x 5 c0 1 c0
c1 x n0
x2
x4
x6
x 2n
1n
2!
4!
6!
2n!
1n
x5
x7
x3
x 2n1
1n
3!
5!
7!
2n 1!
x 2n
x 2n1
c1 1n
2n!
2n 1!
n0
Notice that there are two arbitrary constants, c0 and c1.
NOTE 1
We recognize the series obtained in Example 1 as being the Maclaurin series
for cos x and sin x. (See Equations 8.7.16 and 8.7.15.) Therefore, we could write the solution as
■
yx c0 cos x c1 sin x
But we are not usually able to express power series solutions of differential equations in
terms of known functions.
USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS ■ 3
EXAMPLE 2 Solve y 2xy y 0.
SOLUTION We assume there is a solution of the form
y
cx
n
n
n0
y Then
nc x
n
n1
n1
y and
nn 1c x
n
n2
n2
n 2n 1c
n2
xn
n0
as in Example 1. Substituting in the differential equation, we get
n 2n 1c
x n 2x
n2
n0
n2
xn n0
n
n1
xn n
n
n1
n2
n0
cx
n
n
0
n
0
n0
n 2n 1c
xn
n1
2nc x
2nc
n
n1
n 2n 1c
2nc
nc x
n
cx
n
n0
2n 1cn x n 0
n0
This equation is true if the coefficient of x n is 0:
n 2n 1cn2 2n 1cn 0
7
cn2 2n 1
cn
n 1n 2
n 0, 1, 2, 3, . . .
We solve this recursion relation by putting n 0, 1, 2, 3, . . . successively in Equation 7:
Put n 0:
c2 1
c0
12
Put n 1:
c3 1
c1
23
Put n 2:
c4 3
3
3
c2 c0 c0
34
1234
4!
Put n 3:
c5 5
15
15
c3 c1 c1
45
2345
5!
Put n 4:
c6 7
37
37
c4 c0 c0
56
4! 5 6
6!
Put n 5:
c7 9
159
159
c5 c1 c1
67
5! 6 7
7!
Put n 6:
c8 11
3 7 11
c6 c0
78
8!
Put n 7:
c9 13
1 5 9 13
c7 c1
89
9!
4 ■ USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS
In general, the even coefficients are given by
c2n 3 7 11 4n 5
c0
2n!
and the odd coefficients are given by
1 5 9 4n 3
c1
2n 1!
c2n1 The solution is
y c0 c1 x c2 x 2 c3 x 3 c4 x 4 c0 1 1 2
3 4
37 6
3 7 11 8
x x x x 2!
4!
6!
8!
c1 x 1 3
15 5
159 7
1 5 9 13 9
x x x x 3!
5!
7!
9!
or
y c0 1 8
1 2
3 7 4n 5 2n
x x
2!
2n!
n2
c1 x n1
1 5 9 4n 3 2n1
x
2n 1!
NOTE 2
In Example 2 we had to assume that the differential equation had a series solution. But now we could verify directly that the function given by Equation 8 is indeed a
solution.
NOTE 3
Unlike the situation of Example 1, the power series that arise in the solution of
Example 2 do not define elementary functions. The functions
■
■
2
and
T¸
y1x 1 1 2
3 7 4n 5 2n
x x
2!
2n!
n2
y2x x n1
2
_2
T¡¸
1 5 9 4n 3 2n1
x
2n 1!
are perfectly good functions but they can’t be expressed in terms of familiar functions. We
can use these power series expressions for y1 and y2 to compute approximate values of the
functions and even to graph them. Figure 1 shows the first few partial sums T0, T2, T4, . . .
(Taylor polynomials) for y1x, and we see how they converge to y1 . In this way we can
graph both y1 and y2 in Figure 2.
_8
NOTE 4
FIGURE 1
■
If we were asked to solve the initial-value problem
y 2 xy y 0
y0 0
y0 1
15
we would observe that
fi
_2.5
2.5
›
_15
FIGURE 2
c0 y0 0
c1 y0 1
This would simplify the calculations in Example 2, since all of the even coefficients would
be 0. The solution to the initial-value problem is
yx x n1
1 5 9 4n 3 2n1
x
2n 1!
USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS ■ 5
Exercises
A Click here for answers.
1–11
S
10. y x 2 y 0,
Click here for solutions.
Use power series to solve the differential equation.
1. y y 0
2. y x y
3. y x y
4. x 3y 2y 0
5. y x y y 0
6. y y
2
9. y x y y 0,
y0 0
11. y x y x y 0,
y0 0,
■
■
■
■
■
■
x 2 y x y x 2 y 0
;
y0 1,
y0 0
■
y0 1
■
■
■
■
■
12. The solution of the initial-value problem
7. x 2 1y x y y 0
8. y x y
y0 1,
2
y0 1
y0 0
is called a Bessel function of order 0.
(a) Solve the initial-value problem to find a power series
expansion for the Bessel function.
(b) Graph several Taylor polynomials until you reach one that
looks like a good approximation to the Bessel function on
the interval 5, 5.
■
6 ■ USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS
Answers
Click here for solutions.
S
1.
c0
n0
5. c0
n0
xn
c0 e x
n!
n0
n0
x 3n
3
c0 e x 3
3nn!
1n 2n
2n n! 2n1
x c1 x
n
2 n!
n0 2n 1!
7. c0 c1 x c0
9.
3. c0
x2
1n12n 3! 2n
c0 x
2n2
2
n!n 2!
n2 2
x 2n
2
e x 2
2nn!
11. x n1
1n225 2 3n 12 3n1
x
3n 1!
USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS ■ 7
Solutions: Using Series to Solve Differential Equations
1. Let y(x) =
∞
P
cn xn . Then y 0 (x) =
n=0
∞
P
n=1
so
ncn xn−1 −
∞
P
n=0
∞
P
n=1
∞
P
ncn xn−1 and the given equation, y 0 − y = 0, becomes
cn xn = 0. Replacing n by n + 1 in the first sum gives
n=0
∞
P
(n + 1)cn+1 xn −
n=0
∞
P
cn xn = 0,
n=0
[(n + 1)cn+1 − cn ]xn = 0. Equating coefficients gives (n + 1)cn+1 − cn = 0, so the recursion relation is
cn
1
c0
1
1 1
c0
1
c0
, n = 0, 1, 2, . . . . Then c1 = c0 , c2 = c1 = , c3 = c2 = · c0 = , c4 = c3 = , and
n+1
2
2
3
3 2
3!
4
4!
c0
in general, cn = . Thus, the solution is
n!
∞
∞
∞
X
X
X
c0 n
xn
y(x) =
x = c0
= c0 ex
cn xn =
n!
n!
n=0
n=0
n=0
cn+1 =
3. Assuming y(x) =
∞
P
cn xn , we have y 0 (x) =
n=0
−x2 y = −
∞
P
n=0
∞
P
n=0
ncn xn−1 =
n=1
cn xn+2 = −
(n + 1)cn+1 xn −
∞
P
∞
P
∞
P
∞
P
(n + 1)cn+1 xn and
n=0
cn−2 xn . Hence, the equation y0 = x2 y becomes
n=2
cn−2 xn = 0 or c1 + 2c2 x +
n=2
∞
P
[(n + 1)cn+1 − cn−2 ] xn = 0. Equating coefficients
n=2
cn−2
for n = 2, 3, . . . . But c1 = 0, so c4 = 0 and c7 = 0 and in general
n+1
c0
c0
c0
c3
=
= 2
,
c3n+1 = 0. Similarly c2 = 0 so c3n+2 = 0. Finally c3 = , c6 =
3
6
6·3
3 · 2!
c0
c0
c6
c0
c9 =
=
= 3
, . . . , and c3n = n
. Thus, the solution is
9
9·6·3
3 · 3!
3 · n!
∞
∞
∞
∞
∞ ¡ 3 ¢n
X
X
X
X
X
3
x /3
c0
x3n
n
3n
3n
y (x) =
x = c0
= c0
= c0 ex /3
cn x =
c3n x =
n · n!
n n!
3
3
n!
n=0
n=0
n=0
n=0
n=0
gives c1 = c2 = 0 and cn+1 =
P
P
n−1
n
⇒ y 0 (x) = ∞
and y 00 (x) = ∞
n=1 ncn x
n=0 (n + 2)(n + 1)cn+2 x . The
P
P∞
P
n
n−1
n
differential equation becomes ∞
+ ∞
n=0 (n + 2)(n + 1)cn+2 x + x
n=1 ncn x
n=0 cn x = 0 or
¡
¢
P∞
P∞
P∞
n
n
n
since
. Equating coefficients gives
n=0 [(n + 2)(n + 1)cn+2 + ncn + cn ]x
n=1 ncn x =
n=0 ncn x
5. Let y (x) =
P∞
n
n=0 cn x
cn
−(n + 1)cn
=−
,
(n + 2)(n + 1)
n+2
c0
c2
c4
c0
c0
n = 0, 1, 2, . . . . Then the even coefficients are given by c2 = − , c4 = − =
, c6 = − = −
,
2
4
2·4
6
2·4·6
(n + 2)(n + 1)cn+2 + (n + 1)cn = 0, thus the recursion relation is cn+2 =
and in general, c2n = (−1)n
c7 = −
c0
c1
c3
(−1)n c0
c1
=
. The odd coefficients are c3 = − , c5 = − =
,
2 · 4 · · · · · 2n
2n n!
3
5
3·5
c1
c5
c1
(−2)n n! c1
=−
, and in general, c2n+1 = (−1)n
=
. The solution is
7
3·5·7
3 · 5 · 7 · · · · · (2n + 1)
(2n + 1)!
y (x) = c0
∞
∞
X
X
(−1)n 2n
(−2)n n! 2n+1
+
c
.
x
x
1
2n n!
(2n + 1)!
n=0
n=0
8 ■ USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS
P
P∞
P
n
00
n−2
n
7. Let y (x) = ∞
, xy 0 = ∞
n=0 cn x . Then y =
n=0 n (n − 1) cn x
n=0 ncn x and
¡ 2
¢ 00 P∞
P
n
x + 1 y = n=0 n (n − 1) cn xn + ∞
n=0 (n + 2) (n + 1) cn+2 x . The differential equation becomes
P∞
n=0
[(n + 2) (n + 1) cn+2 + [n (n − 1) + n − 1] cn ] xn = 0. The recursion relation is cn+2 = −
n = 0, 1, 2, . . . . Given c0 and c1 , c2 =
c2n = (−1)n−1
0 · c1
=0
3
n=0
∞
P
y 00 (x) =
⇒
c2n+1 = 0 for n = 1, 2, . . . . Thus the solution is
∞
X
x2
(−1)n−1 (2n − 3)! 2n
+ c0
x .
2
22n−2 n! (n − 2)!
n=2
y (x) = c0 + c1 x + c0
∞
P
3c0
c0
c2
3c4
c0
, c4 = − = − 2
, c6 = −
= (−1)2 3
, ...,
2
4
2 · 2!
6
2 · 3!
1 · 3 · · · · · (2n − 3) c0
(2n − 3)! c0
(2n − 3)! c0
= (−1)n−1 n n−2
= (−1)n−1 2n−2
for
2n n!
2 2
n! (n − 2)!
2
n! (n − 2)!
n = 2, 3, . . . . c3 =
9. Let y(x) =
(n − 1) cn
,
n+2
cn xn . Then −xy 0 (x) = −x
∞
P
n=1
ncn xn−1 = −
∞
P
n=1
ncn xn = −
∞
P
ncn xn ,
n=0
(n + 2)(n + 1)cn+2 xn , and the equation y00 − xy0 − y = 0 becomes
n=0
∞
P
n=0
[(n + 2)(n + 1)cn+2 − ncn − cn ]xn = 0. Thus, the recursion relation is
cn+2 =
ncn + cn
cn (n + 1)
cn
=
=
for n = 0, 1, 2, . . . . One of the given conditions is
(n + 2)(n + 1)
(n + 2)(n + 1)
n+2
y(0) = 1. But y(0) =
∞
X
n=0
cn (0)n = c0 + 0 + 0 + · · · = c0 , so c0 = 1. Hence, c2 =
1
1
c0
c2
= , c4 =
=
,
2
2
4
2·4
c4
1
1
c6 =
=
, . . . , c2n = n . The other given condition is y0 (0) = 0. But
6
2·4·6
2 n!
∞
X
c1
y 0 (0) =
= 0, c5 = 0, . . . ,
ncn (0)n−1 = c1 + 0 + 0 + · · · = c1 , so c1 = 0. By the recursion relation, c3 =
3
n=1
c2n+1 = 0 for n = 0, 1, 2, . . . . Thus, the solution to the initial-value problem is
y(x) =
∞
X
n
cn x =
n=0
11. Assuming that y(x) =
∞
P
∞
X
2n
c2n x
n=0
cn xn , we have xy = x
n=0
x2 y 0 = x2
∞
P
∞
P
n=0
∞
P
ncn xn−1 =
n=1
∞
∞ ¡ 2 ¢n
X
X
2
x /2
x2n
=
=
= e x /2
n n!
2
n!
n=0
n=0
ncn xn+1 ,
cn xn =
∞
P
cn xn+1 ,
n=0
n=0
y 00 (x) =
∞
P
n=2
n(n − 1)cn xn−2 =
= 2c2 +
∞
P
∞
P
(n + 3)(n + 2)cn+3 xn+1
n=−1
(n + 3)(n + 2)cn+3 xn+1 ,
n=0
[replace n with n + 3]
USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS ■ 9
and the equation y00 + x2 y0 + xy = 0 becomes 2c2 +
∞
P
[(n + 3)(n + 2)cn+3 + ncn + cn ] xn+1 = 0.
n=0
So c2 = 0 and the recursion relation is cn+3 =
(n + 1)cn
−ncn − cn
=−
, n = 0, 1, 2, . . . .
(n + 3)(n + 2)
(n + 3)(n + 2)
But c0 = y(0) = 0 = c2 and by the recursion relation, c3n = c3n+2 = 0 for n = 0, 1, 2, . . . .
Also, c1 = y0 (0) = 1, so
c4 = −
2·5
22 52
2c1
5c4
2
=−
, c7 = −
= (−1)2
= (−1)2
, ...,
4·3
4·3
7·6
7·6·4·3
7!
c3n+1 = (−1)n
22 52 · · · · · (3n − 1)2
. Thus, the solution is
(3n + 1)!
y(x) =
∞
X
n=0
cn xn = x +
∞ ·
X
n=1
(−1)n
22 52 · · · · · (3n − 1)2 x3n+1
(3n + 1)!
¸