# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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]T(n + 4)(n + 3)an+2(x — x0)n. (4)

n=0

You can readily verify that the terms in the series (3) and (4) are exactly the same.

236

Chapter 5. Series Solutions of Second Order Linear Equations

EXAMPLE

5

EXAMPLE

6

Write the expression

x2 Y (r + n)anxr+n-1 (5)

n= 0

as a series whose generic term involves xr+n.

First take the x2 inside the summation, obtaining

+ n)anxr +n+1. (6)

n=0

Next, shift the index down by 1 and start counting 1 higher. Thus

J2(r + n)anxr+n+1 = J2 (r + n - 1)an-1 xr+n. (7)

n=0 n=1

Again, you can easily verify that the two series in Eq. (7) are identical, and that both are exactly the same as the expression (5).

Assume that

CO CO

T.na,xn-' = ? a„xn (8)

n=1 n=0

for all x, and determine what this implies about the coefficients an.

We want to use statement 10 to equate corresponding coefficients in the two series. In order to do this, we must first rewrite Eq. (8) so that the series display the same power of x in their generic terms. For instance, in the series on the left side of Eq. (8), we can replace n by n + 1 and start counting 1 lower. Thus Eq. (8) becomes

J2(n + 1)an+1xn = J2 anx ? (9)

n=0 n=0

According to statement 10 we conclude that

(n + 1)an+1 = an, n = 0, 1, 2, 3,...

or

a

an+1 = y+X ’ n = 0 1 2 3,??? (10)

Hence, choosing successive values of n in Eq. (10), we have

a1 a0 a2 a0

a = an, a = — = —, a = — = —,

1 0’ 2 2 2 ’ 3 3 3! ’

and so forth. In general,

an

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

“ xn

Yanx" = aoJ2 nr = aoeX

n=0 n=0 n

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. ? (v - 3)”

n=0

2. ? —xn

2—/ r\n n=0 2

u.2n

3. ? —

n=0 ”!

4. ? 2nxn

5. ?

(2x + 1)n

«= 1 n

6. ?

(x - Xo)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,

xo = 0

xo = 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, 3,....

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

19. ? an (x — 1)n+ = ? an_ 1(x — 1)1

n=0

n=1

20. ? ak+1x + ? akx + = a1 + ? (ak+1 + ak-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 x n .

21. ? n(n — 1)anxn

22. ? a xn+2

n

23. x ? nanxn + ? akx

n=1 k=0

24. (1 — x2) ^ n(n —

n=2 n

n—2

25. m m(m — l)amxm 2 + x ^ ka^x"'

m k=1

.„k—1

26. na xn-1 + x a xn

nn

27. x n— n1n — 1)anxn +

n=2 n

n 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

V'' na xn—1 + 2 'S~'' a xn = 0

nn 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)

d y dy

+Q (x > ddy

dx WA,

since the procedure for the corresponding nonhomogeneous equation is similar.

A wide class of problems in mathematical physics leads to equations of the form (1) having polynomial coefficients; for example, the Bessel equation

x 2y" + xyf + (x2 — v2) y = 0,

where v is a constant, and the Legendre equation

(1 — x 2)y" — 2xy' + a(a + 1)y = 0,

where a is a constant. For this reason, as well as to simplify the algebraic computations, we primarily consider the case in which the functions P, Q, and R are polynomials. However, as we will see, the method of solution is also applicable when P, Q, and R are general analytic functions.

For the present, then, suppose that P, Q, and R are polynomials, and that they have no common factors. Suppose also that we wish to solve Eq. (1) in the neighborhood of a point x0. The solution of Eq. (1) in an interval containing x0 is closely associated with the behavior of P in that interval.

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