Explanation Given: The given differential equation is, 3 x y ″ + 2 ( 1 − x ) y ′ − 4 y = 0 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 ) Where a 0 is arbitrary and the a n ’s are defined in terms of a 0 and r 1 . Calculation: Consider the given equation, 3 x y ″ + 2 ( 1 − x ) y ′ − 4 y = 0 p ( x ) = 2 ( 1 − x ) 3 x and q ( x ) = − 4 3 x x = 0 is a regular singular point as both p ( x ) and q ( x ) are analytic at x = 0 and, p 0 = lim x → 0 x p ( x ) = 2 3 q 0 = lim x → 0 x 2 q ( 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 8.6, Problem 33E

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The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation that is decreasing for x>0.

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Chapter 8.6, Problem 32E

Chapter 8.6, Problem 34E

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Chapter 8 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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To state: The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x = 0 and solution to the given differential equation that is decreasing for x > 0 .

Solution Summary: The author states the recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation.

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To state:

The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation that is decreasing for x>0.

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Chapter 8 Solutions

To state: The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x = 0 and solution to the given differential equation that is decreasing for x &gt; 0 .

Solution Summary: The author states the recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation.

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To state: The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation that is decreasing for x>0.

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Chapter 8 Solutions

To state: The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x = 0 and solution to the given differential equation that is decreasing for x > 0 .

To state:

The recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x=0 and solution to the given differential equation that is decreasing for x>0.