Explanation Given: The differential equation is, w ′ + x w = e x Approach: Assume the differential equation given as, y ″ ( x ) + p ( x ) y ′ ( x ) + q ( x ) y ( x ) = 0 Suppose x 0 is an ordinary point for given differential equation. Then the equation has two linearly independent analytic solutions of the form, y ( x ) = ∑ n = 0 ∞ a n ( x − x 0 ) n Calculation: The differential equation is, w ′ + x w = e x ( 1 ) Let w ( x ) = ∑ n = 0 ∞ a n x n be the solution to the equation ( 1 ) Differentiate w ( x ) with respect to x as, w ′ ( x ) = ∑ n = 0 ∞ n a n x n − 1 The expansion for e x is given as, e x = 1 + x − x 2 2 ! + x 3 3 ! − … Substitute ∑ n = 0 ∞ n a n x n − 1 for w ′ ( x ) and 1 + x − x 2 2 ! + x 3 3 ! − … for e x in equation ( 1 ) as, w ′ + x w = e x ∑ n = 0 ∞ n a n x n − 1 + x ∑ n = 0 ∞ a n x n = 1 + x − x 2 2 ! + x 3 3 ! − … ( [ a 1 + 2 a 2 x + 3 a 3 x 2 + 4 a 4 x 4 … ] + [ a 0 x + a 1 x 2 + a 2 x 3 + a 3 x 4 … ] ) = 1 + x − x 2 2 ! + x 3 3 ! − … ( a 1 + ( 2 a 2 + a 0 ) x + ( 3 a 3 + a 1 ) x 2 + ( 4 a 4 + a 2 ) x 3 + ( 5 a 5 + a 3 ) x 4 + … ) = 1 + x − x 2 2 ! + x 3 3 ! − … Equate the coefficients of like terms on both sides of the above equation as, a 1 = 1

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 8.4, Problem 22E

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The first four nonzero terms in the power series expansion about x=0

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Chapter 8.4, Problem 21E

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