Explanation Given: The differential equation is 16 x 2 y ″ + 32 x y ′ + ( x 4 − 12 ) y = 0 . Formula used: The solution of the differential equation of the form, y ″ + ( 1 − 2 a x ) y ′ + ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) y = 0 , p is not an integer, in terms of Bessel function is given by y = x a [ c 1 J p ( b x c ) + c 2 J − p ( b x c ) ] . Calculation: Consider the differential equation. y ″ + ( 1 − 2 a x ) y ′ + ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) y = 0 ...... ( 1 ) Divide the given equation by 16 x 2 . y ″ + 2 x y ′ + ( 1 16 x 2 − 12 x 2 ) y = 0 ...... ( 2 ) Compare the coefficient of y ′ and y in equation (1) and (2). 2 x = ( 1 − 2 a x ) and 1 16 x 2 − 12 x 2 = ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) The results obtained are as follows: 1 − 2 a = 2 , 2 c − 2 = 2 , b 2 c 2 = 1 16 and a 2 − p 2 c 2 = − 3 4 Further simplify the above equation

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 6.4, Problem 25E

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The general solution of the differential equation 16x2y″+32xy′+(x4−12)y=0 in terms of Bessel function.

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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The general solution of the differential equation 16 x 2 y ″ + 32 x y ′ + ( x 4 − 12 ) y = 0 in terms of Bessel function.

Solution Summary: The author explains the general solution of the differential equation 16x2y′′+32x

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The general solution of the differential equation 16x2y″+32xy′+(x4−12)y=0 in terms of Bessel function.

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