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 form. 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 2 x − 2 ( D 2 + D ) y = sin t , x + D y = 0 . Consider that D 2 x − 2 ( D 2 + D ) y = sin t … ( 1 ) x + D y = 0 … ( 2 ) To eliminate x . Multiply ( 2 ) by D 2 . Subtract ( 1 ) − ( 2 ) . Thus, D 2 x − 2 ( D 2 + D ) y − D 2 x − D 3 y = sin t − 2 ( D 2 + D ) y − D 3 y = sin t ( D 3 + 2 D 2 + 2 D ) y = − sin t Find the roots of auxiliary equation of the homogeneous equation associate to nonhomogeneous DE ( D 3 + 2 D 2 + 2 D ) y = − sin t . Therefore, m 3 + 2 m 2 + 2 m = 0 m ( m 2 + 2 m + 2 ) = 0 m 1 = 0 , m 2 + 2 m + 2 = 0 The root of m 2 + 2 m + 2 = 0 . m = − ( b ) ± b 2 − 4 a c 2 a = − 2 ± 4 − 4 ( 1 ) ( 2 ) 2 ( 1 ) = − 1 ± i Thus, the complementary function is y c = c 1 + e − t ( c 2 cos t + c 3 sin t ) . The right-hand side of the DE is a trigonometric function. Therefore, consider the particular solution in form y p = A cos t + B sin t . Substitute [ y p = A cos t + B sin t . y ′ p = − A sin t + B cos t , y ″ p = − A cos t − B sin t , y ‴ p = A sin t − B cos t ] into the DE. Thus, A sin t − B cos t − 2 A cos t − 2 B sin t − 2 A sin t + 2 B cos t = − sin t sin t ( A − 2 B − 2 A ) + cost ( − B − 2 A + 2 B ) = − sin t ( − 2 B − A ) sin t + ( B − 2 A ) cos t = − sin t Compare, the coefficients of like terms on both sides

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 4.9, Problem 16E

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To solve: The system of differential equations D2x−2(D2+D)y=sint,x+Dy=0 by systematic elimination.

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Chapter 4.9, Problem 15E

Chapter 4.9, Problem 17E

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Differential Equations with Boundary-Value Problems

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The system of differential equations D 2 x − 2 ( D 2 + D ) y = sin t , x + D y = 0 by systematic elimination.

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To solve: The system of differential equations D2x−2(D2+D)y=sint,x+Dy=0 by systematic elimination.

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Chapter 4 Solutions