Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).
(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.

(a) The values are \({j_1}(x) = \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx\) and \({j_2}(x) = \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx\).

(b) The graph has been plotted.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Define Spherical Bessel’s equation.

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

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

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.

Step 2: Find the value of \({j_1}(x)\).

Using the above formula and the result of Problem 32. Let,

\(\begin{aligned}{j_1}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{3/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\sqrt {\frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{2}{{\pi x}}} cosx} \right)\\ &= \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} cosx\\ &= \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx\end{aligned}\)

Step 3: Find the value of \({j_2}(x)\).

Let,

\(\begin{aligned}{j_2}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{5/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\left( {\sqrt {\frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{{18}}{{\pi {x^3}}}} cosx} \right)\\ &= \left( {\sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^3}}}} cosx\\ &= \left( {\sqrt {\frac{9}{{{x^6}}}} - \sqrt {\frac{1}{{{x^2}}}} } \right)sinx - \sqrt {\frac{9}{{{x^4}}}} cosx\\ &= \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx\end{aligned}\)

Step 4: Graph of the Bessel’s function.

Let the graph of \({j_1}(x)\) and \({j_2}(x)\) be,

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).
(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.

Step by Step Solution

Step 1:Define Spherical Bessel’s equation.

Step 2: Find the value of \({j_1}(x)\).

Step 3: Find the value of \({j_2}(x)\).

Step 4: Graph of the Bessel’s function.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.

Step by Step Solution

Step 1:Define Spherical Bessel’s equation.

Step 2: Find the value of \({j_1}(x)\).

Step 3: Find the value of \({j_2}(x)\).

Step 4: Graph of the Bessel’s function.

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(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).
(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.