Explanation Given: The differential equation is 4 x 2 y ″ − 4 x y ′ + ( 16 x 2 + 3 ) 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 differential equation by 4 x 2 . y ″ − 1 x y ′ + ( 4 + 3 4 x 2 ) y = 0 ...... ( 2 ) Compare the coefficient of y ′ and y in equation (1) and (2). − 1 x = ( 1 − 2 a x ) and 4 + 3 4 x 2 = ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) Now the results are obtained as follows: 1 − 2 a = − 1 , 2 c − 2 = 0 , b 2 c 2 = 4 and a 2 − p 2 c 2 = 3 4 Further simplify the above equation. a = 1 , c = 1 , b = 2 and p = 1 2 Substitute the values of above constants in the solution as follows

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Chapter 5.3, Problem 26E

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The general solution of the differential equation 4x2y″−4xy′+(16x2+3)y=0 in terms of Bessel functions.

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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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The general solution of the differential equation 4 x 2 y ″ − 4 x y ′ + ( 16 x 2 + 3 ) y = 0 in terms of Bessel functions.

Solution Summary: The author explains the general solution of the differential equation 4x2y′′-4x

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The general solution of the differential equation 4x2y″−4xy′+(16x2+3)y=0 in terms of Bessel functions.

Chapter 5 Solutions