Explanation Given: The differential equation x 2 y ' ' − x y ' + y = 2 x . It can be written as x 2 d 2 y d x 2 − x d y d x + y = 2 x . Since the equation is nonhomogeneous, first solve the associated homogeneous equation. Calculations: Assume y = x m be a solution of the homogeneous differential equation. Differentiating twice d y d x = m x m − 1 , d 2 y d x 2 = m ( m − 1 ) x m − 2 Substitute y and d 2 y d x 2 into the differential equation yields: x 2 . m ( m − 1 ) x m − 2 − x m x m − 1 + x m = 0 ( m 2 − 2 m + 1 ) x m = 0 Solving the auxiliary equation ( m − 1 ) ( m − 1 ) = 0 ,we get m = 1 with multiplicity 2. The complementary solutions is y c = c 1 x + c 2 x ln x . x 2 y ' ' − x y ' + y = 2 x y ' ' − x − 1 y ' + x − 2 y = 2 x − 1 y ' ' − x − 1 y ' + x − 2 y = 2 x − 1 Here f ( x ) = 2 x − 1

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.6, Problem 21E

To determine

The general solution of the given differential equation using variation of parameters

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Chapter 3.6, Problem 20E

Chapter 3.6, Problem 22E

Chapter 3 Solutions

Advanced Engineering Mathematics

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