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Chapter 13: Q41E (page 589)

Use a CAS to graph \({J_{3/2}}(x),{J_{ - 3/2}}(x),{J_{5/2}}(x),\) and \({J_{ - 5/2}}(x)\).

Short Answer

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The graph has been plotted.

Step by step solution

01

Define Spherical Bessel’s equation.

Bessel functions of half-integral order are used to dene two more important functions:

\(\begin{array}{l}{j_n}(x) = \sqrt {\frac{\pi }{{2x}}} {J_{n + 1/2}}(x)\\{y_n}(x) = \sqrt {\frac{\pi }{{2x}}} {Y_{n + 1/2}}(x)\end{array}\)

The function \({j_n}(x)\) is called the spherical Bessel function of the first kind and \({y_n}(x)\) is the spherical Bessel function of the second kind.

02

Find the graph of \({j_{3/2}}(x)\).

Use GNU Octave to plot the functions.

03

Find the value of \({j_{ - 3/2}}(x)\).

Let,

04

Find the value of \({j_{5/2}}(x)\).

Let,

05

Find the value of \({j_{ - 5/2}}(x)\).

Let,

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Most popular questions from this chapter

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.

\(4{x^2}y'' - 4xy' + \left( {16{x^2} + 3} \right)y = 0\)

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

\({J_0}(x) = {J_{ - 1}}(x) = {J_1}(x)\)

(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 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 \({s^2}\frac{{{d^2}x}}{{d{s^2}}} + s\frac{{dx}}{{ds}} + {s^2}x = 0\).

In Problems \(13 - 20\) use \((20)\) to find the general solution of the given differential equation on \((0,\infty )\).

\(9{x^2}y'' + 9xy' + \left( {{x^6} - 36} \right)y = 0\)
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