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

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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n = 1
generates a multiple of J1, and y2 is only determined up to an additive multiple of J1. In accord with the usual practice we choose c2 = 1/22. Then we obtain
-1
24 -2 (-1)
+1
2
-1
242!
1 + |i +1
(H2 + H)
24 2!
It is possible to show that the solution of the recurrence relation (31) is
(-1)m + 1( Hm + Hm_ 1)
2m 2 m !(m - 1)!
with the understanding that H0 = 0. Thus
m 1 , 2,
y2(x) = -J1(x ) ln X +-------
^ Ζ (-1)m (Hm + Hm-1) X2m
22mm!(m - 1)!
x > 0. (32)
The calculation of y2(x) using the alternative procedure [see Eqs. (19) and (20) of Section 5.7] in which we determine the cn(r2) is slightly easier. In particular the latter procedure yields the general formula for c2m without the necessity of solving a recurrence relation of the form (31) (see Problem 11). In this regard the reader may also wish to compare the calculations of the second solution of Bessels equation of order zero in the text and in Problem 10.
c4 =
5.8 Bessel's Equation
289
The second solution of Eq. (3), the Bessel function of the second kind of order one, Y1, is usually taken to be a certain linear combination of J1 and y. Following Copson (Chapter 1), Y1 is defined as

Y1(x) = -[y(x) + (Y  ln)J1(x)], (33)
where y is defined in Eq. (1). The general solution of Eq. (3) for x > 0 is
y = c1 J1(x) + cY1(x).
Notice that while J1 is analytic at x = 0, the second solution Y1 becomes unbounded in the same manner as 1/x as x ^ 0. The graphs of J1 and Y1 are shown in Figure 5.8.5.
FIGURE 5.8.5 The Bessel functions J1 and Y1.
PROBLEMS In each of Problems 1 through 4 show that the given differential equation has a regular singular
³ point at x = 0, and determine two linearly independent solutions for x > 0.
1. x yw + xy' + xy = 0 . x yw + 3xy; + (1 + x)y = 0
3. x y" + xy' + xy = 0 4. x yw + 4xy; + ( + x)y = 0
5. Find two linearly independent solutions of the Bessel equation of order ,
x2y" + xy' + (x  9)y = 0, x > 0.
6. Show that the Bessel equation of order one-half,
x2y" + xy' + (x  ;1 )y = 0, x > 0,
can be reduced to the equation
v" + v = 0
by the change of dependent variable y = x1/v(x). From this conclude that y1(x) = x1/ cos x and y(x) = x1/ sin x are solutions of the Bessel equation of order one-half.
7. Show directly that the series for J0(x), Eq. (7), converges absolutely for all x.
8. Show directly that the series for J1(x), Eq. (7), converges absolutely for all x and that
J0 (x) =  J1(x).
290
Chapter 5. Series Solutions of Second Order Linear Equations
9. Consider the Bessel equation of order v,
x2y" + xy' + (x2  v2) = 0, x > 0.
Take v real and greater than zero.
(a) Show that x = 0 is a regular singular point, and that the roots of the indicial equation are v and  v.
(b) Corresponding to the larger root v, show that one solution is
Σ³ (x) = xv
1 +jr---------------------^2
1 ς! (1 + v) (2 + v)  (m  1 + v)(m + v) \ 2 /
(c) If 2v is not an integer, show that a second solution is
y2(x) = x-
1 + E----------------------------------------------------------(x )2
m! (1  v) (2  v)    (m  1  v)(m  v) \ 2 /
Note that y1(x) ^ 0 as x ^ 0, and that y2(x) is unbounded as x ^ 0.
(d) Verify by direct methods that the power series in the expressions for y1 (x) and y2(x) converge absolutely for all x. Also verify that y2 is a solution provided only that v is not an integer.
10. In this section we showed that one solution of Bessels equation of order zero,
L [y] = x2 y11 + xy1 + x2 y = 0,
is J0, where J0(x) is given by Eq. (7) with a0 = 1. According to Theorem 5.7.1 a second solution has the form (x > 0)
y2(x) = J0(x) lnx + ^ bnxn.
n=1
(a) Show that
TO
L[y2](x) = ^2n(n  1)bnxn + ^2nbnxn + ^2 bnxn+2 + 2xJ0(x). (i)
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