Explanation Given: The equation x y ″ ( x ) + ( 1 − x ) y ′ ( x ) + n y ( x ) = 0 , Where n is a nonnegative integer, is called Laguerre’s differential equation. Approach: To derive a series solution about the singular point x 0 of the equation a 2 ( x ) y ″ ( x ) + a 1 ( x ) y ′ ( x ) + a 0 ( x ) y ( x ) = 0 , x > x 0 … ( 1 ) The following steps are given below. 1. Set p ( x ) = a 1 ( x ) a 2 ( x ) , q ( x ) = a 0 ( x ) a 2 ( x ) . If both ( x − x 0 ) p ( x ) and ( x − x 0 ) 2 q ( x ) are analytic at x 0 , then x 0 is a regular point and remaining steps apply. 2. Let w ( r , x ) = ( x − x 0 ) r ∑ n = 0 ∞ a n ( x − x 0 ) n = ∑ n = 0 ∞ a n ( x − x 0 ) n + r … ( 2 ) And, using term wise differentiation, substitute w ( r , x ) into equation ( 1 ) to obtain an equation of the form A 0 ( x − x 0 ) r + J + A 1 ( x − x 0 ) r + J + 1 + ... = 0 3. Set the coefficients A 0 , A 1 , A 3 ,… equal to zero. [Notice that the equation A 0 = 0 is just a constant multiple of the indicial equation r ( r − 1 ) + p 0 r + q 0 = 0 here, p 0 = lim x → 0 x p ( x ) , q 0 = lim x → 0 x 2 q ( x ) , and r 0 = lim x → 0 x 3 r ( x ) .] 4. Use the system of equations A 0 = 0 , A 1 = 0 , ... , A k = 0 to find a recurrence relation involving a k and a 0 , a 1 ,…, a k − 1 . 5. Take r = r 1 , the larger root of the indicial equation, and use the obtained in step 4 to determine a 1 , a 2 ,…, recursively in terms of a 0 and r 1 . 6. A series expansion of a solution to equation ( 1 ) is, w ( r 1 , x ) = ( x − x 0 ) r 1 ∑ n = 0 ∞ a n ( x − x 0 ) n , x > x 0 … ( 3 ) Calculation: Consider the equation, x y ″ ( x ) + ( 1 − x ) y ′ ( x ) + n y ( x ) = 0 p ( x ) = 1 − x x and q ( x ) = n x p 0 = lim x → 0 x p ( x ) = lim x → 0 x ( 1 − x x ) = 1 q 0 = lim x → 0 x 2 q ( x ) = lim x → 0 x 2 ( n x ) = 0 The indicial equation is,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.7, Problem 25E

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The series solution of Laguerre’s differential equation for n=0 and n=1.

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Chapter 8.7, Problem 24E

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To state: The series solution of Laguerre’s differential equation for n = 0 and n = 1 .

Solution Summary: The author states the series solution of Laguerre's differential equation for n=0 and N=1.

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The series solution of Laguerre’s differential equation for n=0 and n=1.

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To state: The series solution of Laguerre’s differential equation for n = 0 and n = 1 .

Solution Summary: The author states the series solution of Laguerre's differential equation for n=0 and N=1.

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To state: The series solution of Laguerre’s differential equation for n=0 and n=1.

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Chapter 8 Solutions

To state:

The series solution of Laguerre’s differential equation for n=0 and n=1.