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: The given system of differential equations d 2 x d t 2 = 4 y + e t , d 2 y d t 2 = 4 x − e t . Rewrite the equations in in differential operator notation. D 2 x − 4 y = e t ⋯ ⋯ ( 1 ) D 2 y − 4 x = − e t ⋯ ⋯ ( 2 ) In order to eliminate x , multiply equation (1) by 4 and equation (2) by D 2 and add the resultant equations as follows. 4 D 2 x − 16 y + D 4 y − 4 D 2 x = 4 e t − D 2 e t ( D 4 − 16 ) y = 3 e t Auxiliary equation associated to the homogeneous DE of nonhomogeneous DE ( D 4 − 16 ) y = 3 e t is m 4 − 16 = 0 . Obtain the roots of auxiliary equation as follows. m 4 − 16 = 0 ( m 2 ) 2 − ( 2 2 ) 2 = 0 ( m 2 + 2 ) ( m 2 − 2 ) = 0 m = 2 , − 2 , ± 2 i The roots of the DE are real, unique and conjugated complex roots. Therefore, the complementary function is y c = c 1 e 2 t + c 2 e − 2 t + c 3 cos 2 t + c 4 sin 2 t . In order to find the particular solution, use the method of undermined coefficient. Here, f ( t ) in the DE ( D 4 − 16 ) y = 3 e t is a polynomial function form. So, consider that the particular solution is in form y p = A e t . Substitute y p = A e t , y ′ p = A e t , y ″ p = A e t , y ‴ p = A e t , y p ( 4 ) = A e t in ( D 4 − 16 ) y = 3 e t and obtain the value of A as follows. A e t − 16 A e t = 3 e t − 15 A e t = 3 e t A = − 1 5 Thus, the particular solution is y p = − 1 5 e t

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.12, Problem 7E

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To solve: The system of differential equations d2xdt2=4y+et,d2ydt2=4x−et by systematic elimination.

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Chapter 3.12, Problem 6E

Chapter 3.12, Problem 8E

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Chapter 3 Solutions

Advanced Engineering Mathematics

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

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To solve: The system of differential equations d2xdt2=4y+et,d2ydt2=4x−et by systematic elimination.

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Chapter 3 Solutions