Explanation Given: The given equation is, x 2 y ″ + x y ′ + x 2 y = 0 . Approach: The standard Bessel equation is given as, x 2 y ″ + x y ′ + ( x 2 − v 2 ) y = 0 . The indicial equation for the Bessel equation is given as, r ( r − 1 ) + r − v 2 = 0 . • When the difference of the roots of the indicial equation is not an integer then, there are two linearly independent solutions of Bessel equation are called Bessel functions of the first kind of order v and order − v respectively and they are given as, J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v J − v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 − v + n ) ( x 2 ) 2 n − v • If v is an integer then the other solution of the Bessel equation is called Bessel functions of the second kind of order v and it is given as, Y 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 16E

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A general solution to the given equation using Bessel functions of either the first or second kind.

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A general solution to the given equation using Bessel functions of either the first or second kind.

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