# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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and so forth. In general,

a

an = -0, n= 1, 2,3,.... (11)

n n!

Thus the relation (8) determines all the following coefficients in terms of a0. Finally, using the coefficients given by Eq. (11), we obtain

to xn

J2anx" = a0j2 = a0ex

n=0 n=0 ¦

where we have followed the usual convention that 0! 1.

5.1 Review of Power Series

237

PROBLEMS

In each of Problems 1 through 8 determine the radius of convergence of the given power series.

1. E (x — 3)n

n=0

2. E ~nxn

n

n=0 2

2n

3. E —

n=0 n\

4. 2nxn

5.

(2x + 1)n

n= 1 n

6.

(x — x0)n

n 1 n

7.

n=1

(—1)nn2(x + 2)n 3n

8.

1n

In each of Problems 9 through 16 determine the Taylor series about the point x0 for the given function. Also determine the radius of convergence of the series.

9. sinx, 11. x,

13. ln x,

x0 = 0

x0 = 1

x0 = 1

10. ex 12. x

14.

2

1 + x

x0 = 0 x0 = -1

x0 = 0

15.

1

1x

x0 = 0

16.

1x

x0 = 2

17. Given that y = ^ nxn, compute y' and y" and write out the first four terms of each series

n=0

as well as the coefficient of xn in the general term.

TO

18. Given that y = ^ anxn, compute y' and y" and write out the first four terms of each

n=0

series as well as the coefficient of xn in the general term. Show that if y" = y, then the coefficients a0 and a1 are arbitrary, and determine a2 and a3 in terms of a0 and a1. Show that an+2 = an/(n + 2)(n + 1), n = 0, 1, 2,

In each of Problems 19 and 20 verify the given equation.

19. E an (x — 1)n+ = E a„- 1(x — 1)1

n=0

n=1

20. E ak+1x + E akx + = a1 + E (ak+1 + at-1)

k=0

k=0

k=1

In each of Problems 21 through 27 rewrite the given expression as a sum whose generic term involves xn.

21. Y, n(n — 1)anxn

22. ? a xn+2

n

23. x Y nanxn + E akx

n=1 k=0

24. (1 — x2) ^ n(n — 1')anx

n=2

n—2

25. YI m(m — 1)amxm 2 + x ^ ka,x

m k=1

„k— 1

26. V na xn 1 + xYaxn

nn

27. x n(n — 1)anxn +

n=2

^ anx

n=0

n0

n\ x

n

1

1

n

k

n2

n0

m2

n1

n0

238

Chapter 5. Series Solutions of Second Order Linear Equations

28. Determine the an so that the equation

TO TO

na xn— 1 + 2 a xn = 0

/ j Ï / j Ï

n=1 n=0

TO

is satisfied. Try to identify the function represented by the series ^ anxn.

n=0

5.2 Series Solutions near an Ordinary Point, Part I

In Chapter 3 we described methods of solving second order linear differential equations with constant coefficients. We now consider methods of solving second order linear equations when the coefficients are functions of the independent variable. In this chapter we will denote the independent variable by x. It is sufficient to consider the homogeneous equation

P (x )-+ + Q (x )^~ + R(x) y = 0, (1)

d2 y dy

ldx2 + Q u) ddy

dx ìë

since the procedure for the corresponding nonhomogeneous equation is similar.

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