Explanation Procedure used: General solution of differential equation using systematic elimination: 1. Consider the given set of differential equation L 1 and L 2 . 2. Eliminate any one variable x or y . 3. Find the nonhomogeneous DE and the complementary function x c / y c . 4. Consider a particular solution x p / y p . 5. Find the particular solution and the general solution form x = x c + x p or y = y c + y p 6. Use, the obtained general solution x ( t ) / y ( t ) to find the other one. Calculation: Consider the system of differential equations, ( D − 2 ) x − y = − e t ... ( 1 ) − 3 x + ( D − 4 ) y = − 7 e t ... ( 2 ) Multiply the equation ( 1 ) by ( D − 4 ) and add equation ( 1 ) and ( 2 ) . ( D − 2 ) ( D − 4 ) x − ( D − 4 ) y = 3 e t − 3 x + ( D − 4 ) y = − 7 e t _ ( D − 2 ) ( D − 4 ) x − 3 x = − 4 e t Simplify further. ( D − 2 ) ( D − 4 ) x − 3 x = − 4 e t D 2 x − 2 D x − 4 D x + 8 x − 3 x = − 4 e t ( D 2 − 6 D + 5 ) x = − 4 e t ... ( 3 ) The homogeneous equation associated to DE ( D 2 − 6 D + 5 ) x = − 4 e t is ( D 2 − 6 D + 5 ) x = 0 . Now, the auxiliary equation associated to ( D 2 − 6 D + 5 ) x = 0 is m 2 − 6 m + 5 = 0 . Calculate the roots of auxiliary equation as, m 2 − 6 m + 5 = 0 m 2 − 5 m − m + 5 = 0 m ( m − 5 ) − 1 ( m − 5 ) = 0 ( m − 5 ) ( m − 1 ) = 0 Hence, the roots of the auxiliary equation are m 1 = 1 and m 2 = 5 and the complementary function is x c = c 1 e 5 t + c 2 e t

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 3, Problem 43CR

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To solve: The differential equation (D−2)x−y=−et and −3x+(D−4)y=−7et with the help of systematic elimination.

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Chapter 3, Problem 42CR

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Chapter 3 Solutions

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The differential equation ( D − 2 ) x − y = − e t and − 3 x + ( D − 4 ) y = − 7 e t with the help of systematic elimination.

Solution Summary: The author explains how to solve the differential equation (D-2)x-y=-etand-3x+

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To solve: The differential equation (D−2)x−y=−et and −3x+(D−4)y=−7et with the help of systematic elimination.

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Chapter 3 Solutions