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

**Download**(direct link)

**:**

**442**> 443 444 445 446 447 448 .. 609 >> Next

n=0 n=2 n=2

1

+ 1 2an-1tn = 0,

n-1

n=1

2-1-a2t0 + (3-2-a3 + 2-a0)t1 + [(n+2)(n+1)an+2

n=2

+ (n+1)an-1 + an-2] tn = 0.

It follows that a2 = 0, a3 = -a0/3 and

an+2 = -an-1/(n+2) - an-2/[(n+2)(n+1)], n = 2,3,4... . We

obtain one solution by choosing a1 = 0. Then a4 = -a0/12, a5 = -a2/5 -a1/20 = 0, a6 = -a3/6 - a2/30 = a0/18,... . Thus

one solution is u1(t) = a0(1 - t3/3 - t4/12 + t6/18 + ...) so

y1(x) = u1(x-1) = a0[1 - (x-1)3/3 - (x-1)4/12 + (x-1)6/18 + ...].

00

Section 5.2

79

We obtain a second solution by choosing a0 = 0. Then a4 = -a1/4, a5 = -a2/5 - a1/2 0 = -a1/20,

a6 = -a3/6 - a2/30 = 0, a7 = -a4/7 - a3/42 = a1/28,... .

Thus a second linearly independent solution is

4 5 7

u2(t) = a1[t - t /4 - t /20 + t /28 + ...] or

y2(x) = u2(x-1)

= a1[(x-1) - (x-1) 4/4 - (x-1) 5/20 + (x-1) 7/28 + ...].

2

The Taylor series for x - 1 about x = 1 may be obtained by

22

writing x = (x-1) + 1 so x = (x-1) + 2(x-1) + 1 and

2 2 x - 1 = (x-1) + 2(x-1). The D.E. now appears as

22

y + (x-1) y + [(x-1) + 2(x-1)]y = 0 which is identical to

the transformed equation with t = x - 1.

2 2 2 _ 2 ÷ 2

22b. y = a0 + a1x + a2x + ... , y = a0 + 2a0a1x + (2a0a2 + a1)x

2

+ ... , y = a1 + 2a2x + 3a3x + ... , and

(y') 2 = a1 + 4a1a2x + (6a1a3 + 4a2 2)x2 + ... . Substituting

22

these into (y) = 1 - y and collecting coefficients of

22

like powers of x yields (a1 + a0 - 1) + (4a1a2 + 2a0a1)x +

2 2 2

(6a1a3 + 4a2 + 2a0a2 + a1)x + ... = 0. As in the earlier

problems, each coefficient must be zero. The I.C. y(0) = 0

2 2 2

requires that a0 = 0, and thus a1 + a0 - 1 = 0 gives a1 = 1.

However, the D.E.indicates that y' is always positive, so

y'(0) = a1 > 0 implies a1 = 1. Then 4a1a2 + 2a0a1 = 0 implies

222 that a2 = 0; and 6a1a3 + 4a2 + 2a0a2 + a1 = 6a1a3 + a1 = 0

implies that a3 = -1/6. Thus y = x - x3/3! + ... .

26. We have y(x) = a0y1 + a1y2, where y1 and y2 are found in

80

Section 5.3

Problem 10. Now y(0) = a0 = 0 and y'(0) = a1 = 1. Thus

3 5 7

x x x

y(x) = x - ----- - ----- - ----.

12 240 2240

Section 5.3, Page 253

1. The D.E. can be solved for y" to yield y" = -xy' - y. If y = ôÛ is a solution, then ô"^) = ^ô'Û - ôÛ and thus setting x = 0 we obtain ô"(0) = - 0 - 1 = -1.

Differentiating the equation for y" yields

y''' = -xy" - 2y' and hence setting y = ôÛ again yields ô'''(0) = -0 - 0 = 0. In a similar fashion

ylv = -xy''' - 3y" and thus ô lv(0) = - 0 - 3(-1) = 3. The

process can be continued to calculate higher derivatives of ô(x).

2

6. The zeros of P(x) = x - 2x - 3 are x = -1 and x = 3. For x0 = 4, x0 = -4, and x0 = 0 the distance to the nearest

zero of P(x) is 1,3, and 1, respectively. Thus a lower

bound for the radius of convergence for series solutions in powers of (x-4), (x+4), and x is p = 1, p = 3, and

p = 1, respectively.

9a. Since P(x) = 1 has no zeros, the radius of convergence for x0 = 0 is p = ro.

9f. Since P(x) = x2 + 2 has zeros at x = ±\p2 i, the lower bound for the radius of convergence of the series solution about x0 = 0 is p = .

9h. Since x0 = 1 and P(x) = x has a zero at x = 0, p = 1.

10a. If we assume that y = anxn, then y' = nanxn 1 and

n=2 n=2

ro

n-2

y = Xn(n-1)anx . Substituting in the D.E., shifting

n=2

indices of summation, and collecting coefficients of like powers of x yields the equation

. . 2 0 . . 2 1 (2-1-a, + a a0)x + [3-2-a3 + (a -1)a-,]x

oo

oo

Section 5.3

81

ro

**442**> 443 444 445 446 447 448 .. 609 >> Next