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Chapter 13: Q19E (page 561)
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)
The obtained integral is \(J_0^'(x) = - {J_1}(x) = {J_{ - 1}}(x)\).
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(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).
(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.
In Problems \(23 - 26\) first use \((20)\) to express the general solution of the given differential equation in terms of Bessel functions. Then use \((26)\) and \((27)\) to express the general solution in terms of elementary functions.
\(16{x^2}y'' + 32xy' + \left( {{x^4} - 12} \right)y = 0\)
Use the change of variables \(s = \frac{2}{\alpha }\sqrt {\frac{k}{m}} {e^{ - \alpha t/2}}\) to show that the differential equation of the aging spring \(mx'' + k{e^{ - \alpha t}}x = 0\),\(\alpha > 0\) becomes \({s^2}\frac{{{d^2}x}}{{d{s^2}}} + s\frac{{dx}}{{ds}} + {s^2}x = 0\).
(a) Proceed as in Example \(6\) to show that \(xJ_v^'(x) = - v{J_v}(x) + x{J_{v - 1}}(x)\). (Hint: Write \(2n + v = 2(n + v) - v\).) (b) Use the result in part (a) to derive \((23)\).
Use the formula obtained in Example \(6\) along with part (a) of Problem \(27\) to derive the recurrence relation \(2v{J_v}(x) = x{J_{v + 1}}(x) + x{J_{v - 1}}(x)\).
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