Explanation Given: y ″ − y = 2 t + 4 Approach: Consider a non homogeneous second-order differential equation y ″ + P y ′ + Q y = R Particular solution to the equation using undetermined coefficients is given by y p = A t + B Particular solution to the equation using variation of parameters is given by y p ( t ) = A ( t ) u ( t ) + B ( t ) v ( t ) where u ( t ) , v ( t ) , A ( t ) , and B ( t ) are functions of t and their values are given by following integrals: A ( t ) = − ∫ v ( t ) ⋅ R u v ′ − u ′ v d t B ( t ) = ∫ u ( t ) ⋅ R u v ′ − u ′ v d t Calculation: By undetermined coefficients, the particular solution is y = A t + B y ′ = A y ″ = 0 Putting the values in the given equation 0 − ( A t + B ) = 2 t + 4 A t + B = − 2 t − 4 Therefore, A = − 2 and B = − 4 . y p = − 2 t − 4 By variation of parameters, the particular solution is given by y p ( t ) = A ( t ) u ( t ) + B ( t ) v ( t ) = A ( t ) e t + B ( t ) e − t

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 4.6, Problem 9E

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Chapter 4.6, Problem 8E

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Fundamentals of Differential Equations and Boundary Value Problems

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To find:

The particular solution to the differential equation using undetermined coefficients and variation of parameters and quicker method to solve.