Explanation Procedure used: General solution of differential equation using systematic elimination: 1. Consider the given set of differential equation 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 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 x d t + d y d t = 2 x + 2 y + 1 d x d t + 2 d y d t = y + 3 Simplify the equation d x d t + d y d t = 2 x + 2 y + 1 . d x d t + d y d t = 2 x + 2 y + 1 d x d t + d y d t − 2 x − 2 y = 1 ( D − 2 ) x + ( D − 2 ) y = 1 ... ( 1 ) Simplify the equation d x d t + 2 d y d t = y + 3 . d x d t + 2 d y d t = y + 3 d x d t + 2 d y d t − y = 3 D x + ( 2 D − 1 ) y = 3 ... ( 2 ) Multiply the equation ( 1 ) by D and multiply ( 2 ) by ( D − 2 ) then subtract ( 2 ) from ( 1 ) . D ( D − 2 ) x + D ( D − 2 ) y = 0 [ Since D ( 1 ) = 0 ] D ( D − 2 ) x + ( 2 D − 1 ) ( D − 2 ) y = − 6 _ D ( D − 2 ) y − ( 2 D − 1 ) ( D − 2 ) y = 6 Simplify the above equation. D ( D − 2 ) y − ( 2 D − 1 ) ( D − 2 ) y = 6 D 2 y − 2 D y − 2 D 2 y + D y + 4 D y − 2 y = 6 ( − D 2 y + 3 D y − 2 y ) = 6 ( D 2 − 3 D + 2 ) y = − 6 ... ( 3 ) The homogeneous equation associated to DE ( D 2 − 3 D + 2 ) y = − 6 is ( D 2 − 3 D + 2 ) y = 0 . Now, the auxiliary equation associated to ( D 2 − 3 D + 2 ) y = 0 is m 2 − 3 m + 2 = 0 . Find the roots of the equation m 2 − 3 m + 2 = 0 . m 2 − 3 m + 2 = 0 m 2 − 2 m − m + 2 = 0 m ( m − 2 ) − 1 ( m − 2 ) = 0 ( m − 2 ) ( m − 1 ) = 0 Hence, the roots of the auxiliary equation are m 1 = 2 and m 2 = 1

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 47RE

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To solve: The differential equations dxdt+dydt=2x+2y+1 and dxdt+2dydt=y+3 by the use of systematic elimination.

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Chapter 4 Solutions

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 differential equations d x d t + d y d t = 2 x + 2 y + 1 and d x d t + 2 d y d t = y + 3 by the use of systematic elimination.

Solution Summary: The author explains how to solve differential equations using systematic elimination.

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To solve: The differential equations dxdt+dydt=2x+2y+1 and dxdt+2dydt=y+3 by the use of systematic elimination.

Chapter 4 Solutions