Explanation Procedure used: Solution of system of differential equations by elimination method: 1 Consider the system of differential equations. 2 Express the equations in differential operator notation. 3 Find the DE by elimination. 4 Obtain the complementary function. 5 Further, obtain the particular solution. 6 Ultimately, write the general solution in form y ( x ) = y c + y p . Calculation: Consider the system of differential equations, D x + D 2 y = e 3 t ( 1 ) ( D + 1 ) x + ( D − 1 ) y = 4 e 3 t ( 2 ) To eliminate y , multiply equation (1) by ( D − 1 ) and equation (2) by D 2 and subtract the resultant equation (2) from the resultant equation (1). Thus, ( D − 1 ) D x + D 2 ( D − 1 ) y − D 2 ( D + 1 ) x − D 2 ( D − 1 ) y = − 34 e 3 t ( D − 1 ) D x − D 2 ( D + 1 ) x = − 34 e 3 t ( D 3 + D ) x = 34 e 3 t Auxiliary equation associates homogeneous DE of the nonhomogeneous DE ( D 3 + D ) x = 34 e 3 t is m 3 + m = 0 . Obtain the roots of auxiliary equation as, m 3 + m = 0 m ( m 2 + 1 ) = 0 m 1 = 0 , m 2 = i and m 3 = i Therefore, the complementary function is x c = c 1 + c 2 cos t + c 3 sin t . Consider that the particular solution is x p = A e 3 t . Substitute x p = A e 3 t , x ′ p = 3 A e 3 t , x ″ p = 9 A e 3 t and x ‴ p = 27 A e 3 t into ( D 3 + D ) x = 34 e 3 t . Therefore, 27 A e 3 t + 3 A e 3 t = 34 e 3 t 30 A e 3 t = 34 e 3 t 30 A = 34 A = 17 15 Therefore, the particular solution is x p = 17 15 e 3 t . Therefore, the general solution is x ( t ) = c 1 + c 2 cos t + c 3 sin t + 17 15 e 3 t

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.9, Problem 9E

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To solve: The system of differential equations Dx+D2y=e3t,(D+1)x+(D−1)y=4e3t by systematic elimination.

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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The system of differential equations D x + D 2 y = e 3 t , ( D + 1 ) x + ( D − 1 ) y = 4 e 3 t by systematic elimination.

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To solve: The system of differential equations Dx+D2y=e3t,(D+1)x+(D−1)y=4e3t by systematic elimination.

Chapter 4 Solutions