Explanation Given: From the Exercise 43, the differential equation P ( x ) y ″ + Q ( x ) y ′ + R ( x ) = y the point infinity is an ordinary point if 1 P ( 1 ξ ) [ 2 P ( 1 ξ ) ξ − Q ( 1 ξ ) ξ 2 ] and R ( 1 ξ ) ξ 4 P ( 1 ξ ) have Taylor series expansion about ξ = 0 . That point at infinity is a regular singular point if at least one of the above functions does not have a Taylor series expansion, but both ξ P ( 1 ξ ) [ 2 P ( 1 ξ ) ξ − Q ( 1 ξ ) ξ 2 ] and R ( 1 ξ ) ξ 2 P ( 1 ξ ) do have such expansions. Result used: (1) “Suppose the equation L [ x r ] = x 2 y ″ + α x y ′ + β y = 0 has a solution of the from y = x r , then it becomes L [ x r ] = x r [ r ( r − 1 ) + α r + β ] , where α and β are real constants”. (2) “If r is a root of the quadratic equation F ( r ) = r ( r − 1 ) + α r + β = 0 , then L [ x r ] is zero and y = x r is a solution of x 2 y ″ + α x y ′ + β y = 0 and the roots of the equation are r 1 , r 2 = − ( α − 1 ) ± ( α − 1 ) 2 − 4 β 2 ”

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.4, Problem 44P

To determine

Whether the point at infinity is an ordinary point, a regular singular, or an irregular singular of the differential equation y″+y=0.

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Chapter 5.4, Problem 43P

Chapter 5.4, Problem 45P

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Chapter 5 Solutions

Elementary Differential Equations

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Whether the point at infinity is an ordinary point, a regular singular, or an irregular singular of the differential equation y ″ + y = 0 .

Whether the point at infinity is an ordinary point, a regular singular, or an irregular singular of the differential equation y″+y=0.

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Chapter 5 Solutions