Explanation Given: The given differential equation is, ( x 2 + 2 x − 8 ) y ″ + ( 3 x + 12 ) y ′ − x 2 y = 0 . Approach: A singular point x 0 of y ″ ( x ) + p ( x ) y ′ ( x ) + q ( x ) y ( x ) = 0 is a regular singular point if both ( x − x 0 ) p ( x ) and ( x − x 0 ) 2 q ( x ) are analytic at x 0 .otherwise the point is irregular singular point. If a function is defined at a point x 0 , then it is analytic at the point x 0 . Calculation: Compare the given equation with the standard form to get, p ( x ) = 3 x + 12 x 2 + 2 x − 8 q ( x ) = − x 2 x 2 + 2 x − 8 The singular points are − 4 and 2 . At x 0 = − 4 ( x − x 0 ) p ( x ) = ( x + 4 ) p ( x ) = ( x + 4 ) 3 ( x + 4 ) ( x − 2 ) ( x + 4 ) = 3 ( x + 4 ) x − 2

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 8.6, Problem 9E

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Whether the given differential equation has a regular singular point or an irregular singular point.

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Chapter 8.6, Problem 8E

Chapter 8.6, Problem 10E

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To state: Whether the given differential equation has a regular singular point or an irregular singular point.