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 d t 2 + d y d t = − 5 x ...... ( 1 ) d x d t + d y d t = − x + 4 y ...... ( 2 ) . To eliminate y multiply ( 1 ) by D − 4 and ( 2 ) by D and subtract ( 1 ) − ( 2 ) . ( D 2 + 5 ) ( D − 4 ) x + ( D − 4 ) D y − D ( D + 1 ) x − ( D − 4 ) D y = 0 ( D 2 + 5 ) ( D − 4 ) x − D ( D + 1 ) x = 0 ( D 3 − 5 D 2 + 4 D − 20 ) x = 0 Auxiliary equation associate to the homogeneous DE ( D 3 − 5 D 2 + 4 D − 20 ) x = 0 is m 3 − 5 m 2 + 4 m − 20 = 0 . Find the roots of auxiliary equation. Therefore, m 3 − 5 m 2 + 4 m − 20 = 0 m 2 ( m − 5 ) + 4 ( m − 5 ) = 0 ( m − 5 ) ( m 2 + 4 ) = 0 m 1 = 5 , m 2 = 2 i and m 3 = − 2 i Therefore, the general solution is x ( t ) = c 1 e 5 t + c 2 cos 2 t + c 3 sin 2 t . To eliminate x . Multiply ( 1 ) by ( D + 1 ) and ( 2 ) by ( D 2 + 5 ) . Subtract ( 1 ) − ( 2 ) . ( D 2 + 5 ) ( D + 1 ) x + ( D + 1 ) D y − ( D + 1 ) ( D 2 + 5 ) x − ( D 2 + 5 ) ( D − 4 ) y = 0 ( D + 1 ) D y − ( D 2 + 5 ) ( D − 4 ) y = 0 ( D 3 − 5 D 2 + 4 D − 20 ) y = 0 Auxiliary equation associate to the homogeneous DE ( D 3 − 5 D 2 + 4 D − 20 ) y = 0 is m 3 − 5 m 2 + 4 m − 20 = 0 . Find the roots of auxiliary equation. m 3 − 5 m 2 + 4 m − 20 = 0 m 2 ( m − 5 ) + 4 ( m − 5 ) = 0 ( m − 5 ) ( m 2 + 4 ) = 0 m 1 = 5 , m 2 = 2 i and m 3 = − 2 i Thus, the roots are distinct and two roots are conjugated complex roots. Therefore, the general solution is y ( t ) = c 4 e 5 t + c 5 cos 2 t + c 6 sin 2 t . Substitute, [ x ( t ) = c 1 e 5 t + c 2 cos 2 t + c 3 sin 2 t and y ( t ) = c 4 e 5 t + c 5 cos 2 t + c 6 sin 2 t ] into ( D + 1 ) x + ( D − 4 ) y = 0

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 8E

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To solve: The system of differential equations d2xdt2+dydt=−5x,dxdt+dydt=−x+4y by systematic elimination.

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Chapter 3.12, Problem 7E

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The system of differential equations d 2 x d t 2 + d y d t = − 5 x , d x d t + d y d t = − x + 4 y by systematic elimination.

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To solve: The system of differential equations d2xdt2+dydt=−5x,dxdt+dydt=−x+4y by systematic elimination.

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