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Chapter 13: Q74SE (page 613)

In Problems 29 and 30 use (22) or (23) to obtain the given result.

\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)

Short Answer

Expert verified

The obtained integral is \(\int_0^x r {J_0}(r)dr = x{J_1}(x)\).

Step by step solution

01

Define differential recurrence relation.

Recurrence formulas that relate Bessel functions of different orders are important in theory and in applications.

\(\frac{d}{{dx}}\left( {{x^v}{J_v}(x)} \right) = {x^v}{J_{v - 1}}(x)\) 鈥� (1)

02

Obtain the integration.

Substitute the value\(v = 1\)in the equation (1).

\(x{J_0}(x) = \frac{d}{{dx}}\left( {x{J_1}(x)} \right)\)

Integrate both sides of the expression.

\(\begin{array}{l}\int_0^x r {J_0}(r)dr = \int_0^x {\frac{d}{{dx}}} \left( {x{J_1}(x)} \right)\\\int_0^x r {J_0}(r)dr = x{J_1}(x)\end{array}\)

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Most popular questions from this chapter

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 \(23 - 26\) 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.

\(y'' + y = 0\)

(a) Use the explicit solutions \({y_1}(x)\) and \({y_2}(x)\) of Legendre鈥檚 equation given in \((32)\) and the appropriate choice of \({c_0}\) and \({c_1}\) to find the Legendre polynomials \({P_6}(x)\) and \({P_7}(x)\).

(b) Write the differential equations for which \({P_6}(x)\) and \({P_7}(x)\) are particular solutions.

Show that \(y = {x^{1/2}}w\left( {\frac{2}{3}\alpha {x^{3/2}}} \\(t = \frac{2}{3}\alpha {x^{3/2}}\)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 )

(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.

\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)
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