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 = 2 x + y + t − 2 d y d t = 3 x + 4 y − 4 t From d x d t = 2 x + y + t − 2 , d x d t = 2 x + y + t − 2 d x d t − 2 x − y = t − 2 ( D − 2 ) x − y = t − 2 ... ( 1 ) From d y d t = 3 x + 4 y − 4 t , d y d t = 3 x + 4 y − 4 t d y d t − 4 y − 3 x = − 4 t ( D − 4 ) y − 3 x = − 4 t 3 x − ( D − 4 ) y = 4 t ... ( 2 ) Multiply the equation ( 1 ) by ( D − 4 ) and subtract ( 2 ) from ( 1 ) , ( D − 4 ) ( D − 2 ) x − ( D − 4 ) y = ( D − 4 ) ( t − 2 ) 3 x − ( D − 4 ) y = 4 t _ ( D − 4 ) ( D − 2 ) x − 3 x = ( D − 4 ) ( t − 2 ) − ( 4 t ) Simplify the above equation. ( D − 4 ) ( D − 2 ) x − 3 x = ( D − 4 ) ( t − 2 ) − ( 4 t ) D 2 x − 4 D x − 2 D x + 8 x − 3 x = D t − 4 t − 2 D + 8 − 4 t D 2 x − 6 D x + 5 x = 9 − 8 t ( D 2 − 6 D + 5 ) x = 9 − 8 t ... ( 3 ) The homogeneous equation associated to DE ( D 2 − 6 D + 5 ) x = 9 − 8 t is ( D 2 − 6 D + 5 ) x = 0 . Now, the auxiliary equation associated to ( D 2 − 6 D + 5 ) x = 0 is m 2 − 6 m + 5 = 0

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4, Problem 48RE

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To solve: The differential equation dxdt=2x+y+t−2 and dydt=3x+4y−4t with the help of systematic elimination.

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Chapter 4, Problem 47RE

Chapter 4, Problem 49RE

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A First Course in Differential Equations with Modeling Applications (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 - Given that x(t) = c1 cos t + c2 sin t is the...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 equation d x d t = 2 x + y + t − 2 and d y d t = 3 x + 4 y − 4 t with the help of systematic elimination.

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

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To solve: The differential equation dxdt=2x+y+t−2 and dydt=3x+4y−4t with the help of systematic elimination.

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Chapter 4 Solutions