Explanation Given: The Bessel equation of order one-half t 2 y ′′ + t y ′ + ( t 2 − 1 4 ) y = 0 t > 0 has two linearly independent solutions, y 1 ( t ) = t − 1 / 2 cos t y 2 ( t ) = t − 1 / 2 sin t For nonhomogeneous equation t 2 y ′′ + t y ′ + ( t 2 − 1 4 ) y = t 5 / 2 t > 0 . Approach: The differential equation t 2 y ′′ + t y ′ + ( t 2 − 1 4 ) y = t 5 / 2 y ′′ + 1 t y ′ + ( 1 − 1 4 t 2 ) y = t 1 / 2 ( 1 ) is non-homogeneous and in the standard form, i.e. y ′′ ( t ) + p ( t ) y ′ + q ( t ) y ( t ) = g ( t ) p ( t ) = 1 t q ( t ) = ( 1 − 1 4 t 2 ) g ( t ) = t 1 / 2 . The general solution to (1) is given by y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + Y p ( t ) Y p ( t ) = v 1 ( t ) y 1 ( t ) + v 2 ( t ) y 2 ( t ) ( 2 ) Such that v 1 and v 2 satisfies the following equations y 1 v ′ 1 + y 2 v ′ 2 = 0 y ′ 1 v ′ 1 + y ′ 2 v ′ 2 = g ( t ) ( 3 ) Y p ( t ) is a particular solution to (1) in the standard form solved by the method of variation of parameters, v 1 ( t ) and v 2 ( t ) are unknown functions of t , g ( t ) is a function of t in (1), and c 1 and c 2 are constants. Calculation: Given, y 1 ( t ) = t − 1 / 2 cos t y 2 ( t ) = t − 1 / 2 sin t Substitute y 1 ( t ) and y 2 ( t ) in Y p ( t ) in (2) Y p ( t ) = t − 1 / 2 cos ( t ) v 1 ( t ) + t − 1 / 2 sin ( t ) v 2 ( t ) (4) From (3) v 1 and v 2 should satisfy the following equations y 1 v ′ 1 + y 2 v ′ 2 = 0 y ′ 1 v ′ 1 + y ′ 2 v ′ 2 = g ( t ) Substitute t − 1 / 2 cos t for y 1 , t − 1 / 2 sin t for y 2 , − cos t − 2 t sin t 2 t 3 / 2 for y ′ 1 , − sin t + 2 t cos t 2 t 3 / 2 for y ′ 2 , and t 1 / 2 for g ( t ) in (3)

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 4.7, Problem 40E

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A general solution to the given nonhomogeneous equation.

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Fundamentals of Differential Equations and Boundary Value Problems

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A general solution to the given nonhomogeneous equation.

Solution Summary: The author explains that the Bessel equation of order one-half rllt2y

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To find:

A general solution to the given nonhomogeneous equation.