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Chapter 13: Q15E (page 550)

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

: 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 \(x\).

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 \(23 - 26\) 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.

\(16{x^2}y'' + 32xy' + \left( {{x^4} - 12} \right)y = 0\)

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 \(23 - 26\) 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.

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