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 = 2 x − y , d y d t = x . Write the set of differential equations in differential operator notation. Therefore, ( D − 2 ) x + y = 0 … ( 1 ) x − D y = 0 … ( 2 ) Multiply equation (1) by differential operator D and add with equation (2). Therefore, D [ ( D − 2 ) x + y ] + x − D y = 0 ( D 2 − 2 D ) x + x = 0 x ( D 2 − 2 D + 1 ) = 0 Thus, the characteristic equation is D 2 − 2 D + 1 = 0 . Solve the auxiliary equation for D 2 − 2 D + 1 = 0 . Therefore, m 2 − 2 m + 1 = 0 ( m − 1 ) 2 = 0 m = 1 , 1 Thus, the roots of the DE are equal and repeated. Therefore, the general solution is x ( t ) = c 1 e t + c 2 t e t . Now, eliminate x . Multiply equation (2) by ( D − 2 ) and then subtract from equation (1). Therefore, ( D − 2 ) x + y − ( D − 2 ) ( x − D y ) = 0 y + D 2 y − 2 D y = 0 D 2 y − 2 D y + y = 0 y ( D 2 − 2 D + 1 ) = 0 The characteristic equation is D 2 − 2 D + 1 = 0 . Solve the associated auxiliary equation. Therefore, m 2 − 2 m + 1 = 0 ( m − 1 ) 2 = 0 m = 1 , 1 Thus, the roots are repeated and equal

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 3.12, Problem 1E

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Chapter 3 Solutions

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The system of differential equations d x d t = 2 x − y , d y d t = x by systematic elimination.

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Chapter 3 Solutions