(a) Explanation Formula used: Spherical Bessel function of first kind: j n ( x ) = π 2 x J n + 1 / 2 ( x ) Recurrence relation: (i) 2 ν J ν ( x ) = x J ν + 1 ( x ) + x J ν − 1 ( x ) (ii) x J ′ ν ( x ) = − ν J ν ( x ) + x J ν − 1 ( x ) Bessel functions of order 1 2 and − 1 2 : J 1 / 2 ( x ) = 2 π x sin x and J − 1 / 2 ( x ) = 2 π x cos x . Calculation: Consider the spherical Bessel function of first kind. j n ( x ) = π 2 x J n + 1 / 2 ( x ) Substitute n = 1 and n = 2 in the above Bessel’s function. n = 1 , j 1 ( x ) = π 2 x J 3 / 2 ( x ) ...... ( 1 ) n = 2 , j 2 ( x ) = π 2 x J 5 / 2 ( x ) ...... ( 2 ) Now, use the recurrence relation (i) with ν = 1 / 2 . J 1 / 2 ( x ) = x J 3 / 2 ( x ) + x J − 1 / 2 ( x ) x J 3 / 2 ( x ) = J 1 / 2 ( x ) − x J − 1 / 2 ( x ) J 3 / 2 ( x ) = 1 x J 1 / 2 ( x ) − J − 1 / 2 ( x ) The value of J 3 / 2 ( x ) is calculated as follows. J 3 / 2 ( x ) = 1 x 2 π x sin x − 2 π x cos x = 2 π x ( sin x x − cos x ) Similarly, use the recurrence relation (i) with ν = 3 / 2 . 3 J 3 / 2 ( x ) = x J 5 / 2 ( x ) + x J 1 / 2 ( x ) x J 5 / 2 ( x ) = 3 J 3 / 2 ( x ) − x J 1 / 2 ( x ) J 5 / 2 ( x ) = 3 x J 3 / 2 ( x ) − J 1 / 2 ( x ) The value of J 3 / 2 ( x ) is obtained above (b) To determine To sketch: The graph of j 1 ( x ) and j 2 ( x ) in the same coordinate plane using graphing calculator.

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 6.4, Problem 33E

(a)

To determine

The spherical Bessel functions j1(x) and j2(x).

(b)

To determine

To sketch: The graph of j1(x) and j2(x) in the same coordinate plane using graphing calculator.

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Chapter 6.4, Problem 32E

Chapter 6.4, Problem 34E

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Chapter 6 Solutions

Differential Equations with Boundary-Value Problems

Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 13E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 16E Ch. 6.1 - In Problems 17 and 18 use an appropriate series in...Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - Prob. 28E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - In Problems 3538 proceed as in Example 4 and find...Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.2 - In Problems 1 and 2 without actually solving the...Ch. 6.2 - Prob. 2E Ch. 6.2 - In Problems 3–6 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - Prob. 9E Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Prob. 13E Ch. 6.2 - Prob. 14E Ch. 6.2 - Prob. 15E Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Prob. 23E Ch. 6.2 - In Problems 23 and 24 use the procedure in Example...Ch. 6.2 - Without actually solving the differential equation...Ch. 6.2 - How can the power series method be used to solve...Ch. 6.2 - Prob. 27E Ch. 6.2 - Prob. 28E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 9E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 11E Ch. 6.3 - Prob. 12E Ch. 6.3 - In Problems 13 and 14, x = 0 is a regular singular...Ch. 6.3 - Prob. 14E Ch. 6.3 - In Problems 1524, x = 0 is a regular singular...Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Prob. 23E Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - In Problems 1320 use (20) to find the general...Ch. 6.4 - Prob. 21E Ch. 6.4 - Assume that b in equation (20) can be pure...Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Use the recurrence relation in Problem 28 along...Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Use the change of variables s=2kmet/2 to show that...Ch. 6.4 - Prob. 36E Ch. 6.4 - Use the result in parts (a) and (b) of Problem 36...Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - (a) Use the explicit solutions y1(x) and y2(x) of...Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Find the first three positive values of for which...Ch. 6.4 - The differential equation y 2xy + 2y = 0 is known...Ch. 6.4 - (a) When = n is a nonnegative integer, Hermites...Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6 - In Problems 1 and 2 answer true or false without...Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Express the general solution of the given...Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE

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The spherical Bessel functions j 1 ( x ) and j 2 ( x ) .

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