Page 319 - Schaum's Outline of Differential Equations

302 GAMMA AND BESSEL FUNCTIONS [CHAP. 30

Multiplying the numerator and denominator in the second summation by 2(p + k) and noting that (p + k)T(p + k)
= T(p + k + 1), we find

30.16. Use Problems 30.14 and 30.15 to derive the recurrence formula

Subtracting the results of Problem 30.15 from the results of Problem 30.14, we find that

Upon solving for J p+1(x), we obtain the desired result.

2
30.17. Show that y = xJ v(x) is a solution of xy"-y' -x Jo(x) = 0.
First note that J\(x) is a solution of Bessel's equation of order one:

Now substitute y = xJ^x) into the left side of the given differential equation:

But JQ(X) = -Ji(x) (by (_/) of Problem 30.14), so that the right-hand side becomes

the last equality following from (_/).

2
2
30.18. Show that y = JxJ 3l2(x) is a solution of x y" + (x - 2)y = 0.
Observe that J^^x) is a solution of Bessel's equation of order |;

Now substitute y = iJxJ 3l2(x) into the left side of the given differential equation, obtaining

the last equality following from (_/). Thus *JxJ 3/2(x) satisfies the given differential equation.