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Chapter 13: Q68SE (page 612)
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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In Problems 29 and 30 use (22) or (23) to obtain the given result.
\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)
Show that \(y = {x^{1/2}}w\left( {\frac{2}{3}\alpha {x^{3/2}}} \right)\) is a solution of the given form of Airy鈥檚 differential equation whenever w is a solution of the indicated Bessel鈥檚 equation. (Hint: After differentiating, substituting, and simplifying, then let \(t = \frac{2}{3}\alpha {x^{3/2}}\))
(a) \(y'' + {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' + \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)
(b) \(y'' - {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' - \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)
Assume that b in equation (20) can be pure imaginary, that is, . Use this assumption to express the general solution of the given differential equation in terms of the modified Bessel functions In and Kn.
(a) y0 2 x2y 5 0
(b) xy0 1 y9 2 7x3
(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 \((23)\).
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