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 t = 4 x + 7 y , d y d t = x − 2 y . Write the set of differential equations in differential operator notation. Therefore, ( D − 4 ) x − 7 y = 0 ⋯ ( 1 ) ( D + 2 ) y − x = 0 ⋯ ( 2 ) To eliminate x multiply (2) by ( D − 4 ) and add with equation (1). Therefore, ( D − 4 ) x − 7 y + ( D + 2 ) ( D − 4 ) y − ( D − 4 ) x = 0 ( D 2 + 2 D − 4 D − 8 ) y − 7 y = 0 ( D 2 − 2 D − 15 ) y = 0 Thus, the characteristic equation is D 2 − 2 D − 15 = 0 . Solve the auxiliary equation for D 2 − 2 D − 15 = 0 . Therefore, m 2 − 2 m − 15 = 0 m 2 − 5 m + 3 m − 15 = 0 m ( m − 5 ) + 3 ( m − 5 ) = 0 ( m − 5 ) ( m + 3 ) = 0 Hence, m 1 = 5 and m 2 = − 3 . Thus, the roots of the DE are real and distinct. Therefore, the general solution is y ( t ) = c 1 e 5 t + c 2 e − 3 t . Now, eliminate y . Multiply equation (1) by ( D + 2 ) and then add with equation (2). Therefore, ( D − 4 ) ( D + 2 ) − 7 ( D + 2 ) y + ( D + 2 ) 7 y − 7 x = 0 ( D 2 − 4 D + 2 D − 8 ) x − 7 x = 0 ( D 2 − 2 D − 15 ) x = 0 The characteristic equation is D 2 − 2 D − 15 = 0 . Solve the associated auxiliary equation. Therefore, m 2 − 2 m − 15 = 0 m 2 − 5 m + 3 m − 15 = 0 m ( m − 5 ) + 3 ( m − 5 ) = 0 ( m − 5 ) ( m + 3 ) = 0 Hence, the solutions are m 1 = 5 and m 2 = − 3

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 2E

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To solve: The system of differential equations dxdt=4x+7y,dydt=x−2y 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 t = 4 x + 7 y , d y d t = x − 2 y by systematic elimination.

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To solve: The system of differential equations dxdt=4x+7y,dydt=x−2y by systematic elimination.

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