Explanation Procedure used: General solution of differential equation using systematic elimination: 1. Consider the given set of differential equations L 1 and L 2 . 2. Eliminate any one variable x or y . 3. Find the nonhomogeneous DE and the complementary function x c / y c . 4. Consider a particular solution x p / y p . 5. Find the particular solution and then the general solution form x = x c + x p or y = y c + y p 6. Use the obtained general solution x ( t ) / y ( t ) to find the other one. Calculation: Consider the system of differential equations. ( D + 2 ) x + ( D + 1 ) y = sin 2 t ... ( 1 ) 5 x + ( D + 3 ) y = cos 2 t ... ( 2 ) Multiply the equation ( 1 ) by 5 and ( 2 ) by ( D + 2 ) , then subtract it. ( D + 2 ) 5 x + 5 ( D + 1 ) y = 5 sin 2 t ( D + 2 ) 5 x + ( D + 2 ) ( D + 3 ) y = ( D + 2 ) cos 2 t _ 5 ( D + 1 ) y − ( D + 2 ) ( D + 3 ) y = 5 sin 2 t − ( D + 2 ) cos 2 t Simplify further. 5 ( D + 1 ) y − ( D + 2 ) ( D + 3 ) y = 5 sin 2 t − ( D + 2 ) cos 2 t 5 D y + 5 y − D 2 y − 2 D y − 3 D y − 6 y = 5 sin 2 t + 2 sin 2 t − 2 cos 2 t ( D 2 + 1 ) y = 2 cos 2 t − 7 sin 2 t ... ( 3 ) The homogeneous equation associated to DE ( D 2 + 1 ) y = 2 cos 2 t − 7 sin 2 t is ( D 2 + 1 ) y = 0 . Now, the auxiliary equation associated to ( D 2 + 1 ) y = 0 is m 2 + 1 = 0 . Calculate the roots of auxiliary equation. m 2 + 1 = 0 m 2 = − 1 m = ± i Hence, the roots of the auxiliary equation are m 1 = i and m 2 = − i . Thus, the complementary function is y c = c 1 cos t + c 2 sin t . Let, the particular solution to the DE be y p = A cos 2 t + B sin 2 t . Substitute y p = A cos 2 t + B sin 2 t , y ′ p = − 2 A sin 2 t + 2 B cos 2 t and y ″ p = − 4 A cos 2 t − 4 B sin 2 t into equation ( 3 ) , − 3 A cos 2 t − 3 B sin 2 t = 2 cos 2 t − 7 sin 2 t Compare the coefficients of similar term on both sides, − 3 A = 2 which implies A = − 2 3 − 3 B = − 7 which implies B = 7 3 Thus, the particular solution is y p = − 2 3 cos 2 t + 7 3 sin 2 t

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, Problem 50RE

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To solve: The system of differential equations (D+2)x+(D+1)y=sin2t and 5x+(D+3)y=cos2t by the use of systematic elimination.

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The system of differential equations ( D + 2 ) x + ( D + 1 ) y = sin 2 t and 5 x + ( D + 3 ) y = cos 2 t by the use of systematic elimination.

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To solve: The system of differential equations (D+2)x+(D+1)y=sin2t and 5x+(D+3)y=cos2t by the use of systematic elimination.

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