Explanation Approach: The solution to the Bessel equation of first kind of order v is given as, J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v . Γ ( n + 1 ) = n Γ ( n ) . Calculation: Consider the solution to the Bessel equation of first kind of order v . J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v For the order ( v − 1 ) , the above solution becomes, J v − 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v − 1 + n ) ( x 2 ) 2 n + v − 1 = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v − 1 And for the order ( v + 1 ) the solution is, J v + 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + 1 + n ) ( x 2 ) 2 n + v + 1 = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 2 + v + n ) ( x 2 ) 2 n + v + 1 J v + 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 2 + v + n ) ( x 2 ) 2 n + v + 1 ... ( 1 ) Evaluate the expression as follows, 2 v x J v ( x ) − J v − 1 ( x ) = 2 v x ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v − ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v − 1 = v ∑ n = 0 ∞ ( − 1 ) n n ! ( v + n ) Γ ( v + n ) ( x 2 ) 2 n + v − 1 − ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v x

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 8.8, Problem 23E

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The recurrence formula (33).

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Chapter 8.8, Problem 22E

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To derive: The recurrence formula (33) .

Solution Summary: The author explains how the recurrence formula (33) is derived from the Bessel equation of first kind of order v.

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To derive:

The recurrence formula (33).

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Chapter 8 Solutions

To derive: The recurrence formula (33) .

Solution Summary: The author explains how the recurrence formula (33) is derived from the Bessel equation of first kind of order v.

Videos

To derive: The recurrence formula (33).

Want to see the full answer?

Chapter 8 Solutions

To derive:

The recurrence formula (33).