Explanation Given: The given statement is, “Because x = 0 is an irregular singular point of x 3 y ″ − x y ' + y = 0 , the DE possesses no solution that is analytic at x = 0 ”. Result used: Consider the linear differential equation, y ″ + P ( x ) y ' + Q ( x ) y = 0 (1) A point x = x 0 is called an ordinary point if both P ( x ) and Q ( x ) are analytic at x 0 . If a point is not ordinary, then it is called a singular point. A singular point x = x 0 is said to be regular singular point of the above differential equation if the function p ( x ) = ( x − x 0 ) P ( x ) and q ( x ) = ( x − x 0 ) 2 Q ( x ) are both analytic at x 0 . A singular point that is not regular is said to be an irregular singular point of the equation. Calculation: The given differential equation is, x 3 y ″ − x y ' + y = 0 Divide the above differential equation by, x 3 on both sides. y ″ − 1 x 2 y ' + 1 x 3 y = 0 Comparing the above differential equation with equation (1) it gives, P ( x ) = − 1 x 2 , Q ( x ) = 1 x 3 Observe from above that P ( x ) and Q ( x ) is not analytic at x = 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Advanced Engineering Mathematics

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E

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