James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1180 Chapter 17 Second-Order Differential Equations
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
y 1 sxd − 1 2 1 2! x 3 7 ∙ ∙ ∙ s4n 2 5d
2 2 ò
x 2n
n−2 s2nd!
2
T¸
and
1 5 9 ∙ ∙ ∙ s4n 2 3d
y 2 sxd − x 1 ò
x 2n11
n−1 s2n 1 1d!
_2 2
T¡¸
_8
FIGURE 1
15
fi
_2.5 2.5
›
are perfectly good functions but they can’t be expressed in terms of familiar functions.
We can use these power series expressions for y 1 and y 2 to compute approximate values
of the functions and even to graph them. Figure 1 shows the first few partial sums
T 0 , T 2 , T 4 , . . . (Taylor polynomials) for y 1 sxd, and we see how they converge to y 1 . In this
way we can graph both y 1 and y 2 as in Figure 2.
Note 4 If we were asked to solve the initial-value problem
y0 2 2xy9 1 y − 0 ys0d − 0 y9s0d − 1
we would observe from Theorem 11.10.5 that
c 0 − ys0d − 0 c 1 − y9s0d − 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
FIGURE 2
_15
1 ? 5 ? 9 ? ∙ ∙ ∙ ? s4n 2 3d
ysxd − x 1 ò
x 2n11
n−1 s2n 1 1d!
1–11 Use power series to solve the differential equation.
1. y9 2 y − 0 2. y9 − xy
3. y9 − x 2 y 4. sx 2 3dy9 1 2y − 0
5. y0 1 xy9 1 y − 0 6. y0 − y
7. sx 2 1dy0 1 y9 − 0
8. y0 − xy
9. y0 2 xy9 2 y − 0, ys0d − 1, y9s0d − 0
10. y0 1 x 2 y − 0, ys0d − 1, y9s0d − 0
;
11. y0 1 x 2 y9 1 xy − 0, ys0d − 0, y9s0d − 1
12. The solution of the initial-value problem
x 2 y0 1 xy9 1 x 2 y − 0 ys0d − 1 y9s0d − 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 f25, 5g.
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