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Chapter 14: Q27E (page 647)

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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Step by step solution

01

Define Spherical Bessel’s equation.

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

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

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.

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

(a) Use the general solution obtained in Problem 37 to solve the IVP \[4x'' + tx = 0,x(0.1) = 1,x'(0.1) = - \frac{1}{2}\]. Use a CAS to evaluate coefficients.

(b) Use a CAS to graph the solution obtained in part (a) for \[0 \le t \le 200\].

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 \(x\).

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

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.

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