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Chapter 13: Q58E (page 608)
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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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.
\({x^2}y'' + 4xy' + \left( {{x^2} + 2} \right)y = 0\)
(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 \((23)\).
(a) Use (20) to show that the general solution of the differential equation \(xy'' + \lambda y = 0\) on the interval \((0,\infty )\) is \(y = {c_1}\sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right) + {c_2}\sqrt x {Y_1}\left( {2\sqrt {\lambda x} } \right)\).
(b) Verify by direct substitution that \(y = \sqrt x {J_1}\left( {2\sqrt {\lambda x} } \right)\) is a particular solution of the DE in the case \(\lambda = 1\).
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)
Use the recurrence relation in Problem 28 along with (26) and (27) to express \({J_{3/2}}(x),{J_{5/2}}(x),{J_{ - 3/2}}(x),{J_{ - 5/2}}(x)\) in terms of \(sinx,cosx\), and powers of \(x\).
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