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Chapter 13: Q51E (page 593)
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)
The obtained integral is \(\int_0^x r {J_0}(r)dr = x{J_1}(x)\).
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(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 \(2v{J_v}(x) = x{J_{v + 1}}(x) + x{J_{v - 1}}(x)\).
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) 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.
Proceed as on page \(269\) to derive the elementary form of \({J_{ - 1/2}}(x)\) given in \((27)\).
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