Program Explanation Given information: The linear system of the differential equation x ′ = y + z + e − t , y ′ = x + z , z ′ = x + y . Calculation: The given linear system of the differential equation x ′ = y + z + e − t , y ′ = x + z , z ′ = x + y . Write the system of differential equation in homogeneous equation. x ′ − y − z − e − t = 0 y ′ − x − z = 0 z ′ − x − y = 0 Now write the above equations in the form of D as, D x − y − z = e − t − x + D y − z = 0 − x − y + D z = 0 The resultant equation represents in the form. L 11 x + L 12 y + L 13 z = f 1 ( t ) L 21 x + L 22 y + L 23 z = f 2 ( t ) L 31 x + L 32 y + L 33 = f 3 ( t ) The characteristic equation of the given system of the equation can be found by the determinant method. The system of equation L 11 x + L 12 y + L 13 z = f 1 ( t ) L 21 x + L 22 y + L 23 z = f 2 ( t ) L 31 x + L 32 y + L 33 = f 3 ( t ) can be written in the form of determinant as, | D − 1 − 1 − 1 D − 1 − 1 − 1 D | x = | e − t − 1 − 1 0 D − 1 0 − 1 D | [ D ( D 2 − 1 ) + 1 ( − D − 1 ) − 1 ( 1 + D ) ] x = ( D 2 − 1 ) e − t [ D 3 − 3 D − 2 ] x = 0 Now the corresponding characteristic equation is as follows, r 3 − 3 r − 2 = 0 ( r − 2 ) ( r + 1 ) 2 = 0 Therefore, the roots of the equations are r 1 = 2 , r 2 = − i , r 3 = i . The solution of the differential equation can be written as, x ( t ) = c 1 e 2 t + c 2 e − t + c 3 t e − t Now apply the same method to find the value of y ( t ) , z ( t ) . | D − 1 − 1 − 1 D − 1 − 1 − 1 D | x = | D e − t − 1 − 1 0 − 1 − 1 0 D | [ D ( D 2 − 1 ) + 1 ( − D − 1 ) − 1 ( 1 + D ) ] y = ( − D − 1 ) e − t [ D 3 − 3 D − 2 ] y = 0 The solution of the differential equation can be written as, y ( t ) = d 1 e 2 t + d 2 e − t + d 3 t e − t Now apply the same method to find the value of y ( t ) , z ( t ) . | D − 1 − 1 − 1 D − 1 − 1 − 1 D | x = | D − 1 e − t − 1 0 − 1 − 1 0 D | [ D ( D 2 − 1 ) + 1 ( − D − 1 ) − 1 ( 1 + D ) ] y = ( 1 + D ) e − t [ D 3 − 3 D − 2 ] y = 0 The solution of the differential equation can be written as, z ( t ) = e 1 e 2 t + e 2 e − t + e 3 t e − t Differentiate x ( t ) = c 1 e 2 t + c 2 e − t + c 3 t e − t with respect to t both sides. x ′ ( t ) = ( c 2 − c 1 ) e − t − c 2 t e − t + 2 c 3 e 2 t Differentiate y ( t ) = d 1 e 2 t + d 2 e − t + d 3 t e − t with respect to t both sides

Pearson eText for Differential Equations: Computing and Modeling Tech Update -- Instant Access (Pearson+)

5th Edition

ISBN: 9780137442782

Author: C. Edwards, David Penney

Publisher: PEARSON+

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 4.2, Problem 20P

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x′=y+z+e−t,y′=x+z, z′=x+y .

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Chapter 4.2, Problem 19P

Chapter 4.2, Problem 21P

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Chapter 4 Solutions

Pearson eText for Differential Equations: Computing and Modeling Tech Update -- Instant Access (Pearson+)

Ch. 4.1 - Prob. 1P Ch. 4.1 - Prob. 2P Ch. 4.1 - Prob. 3P Ch. 4.1 - Prob. 4P Ch. 4.1 - Prob. 5P Ch. 4.1 - Prob. 6P Ch. 4.1 - Prob. 7P Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Prob. 10P

Ch. 4.1 - Prob. 11P Ch. 4.1 - Prob. 12P Ch. 4.1 - Prob. 13P Ch. 4.1 - Prob. 14P Ch. 4.1 - Prob. 15P Ch. 4.1 - Prob. 16P Ch. 4.1 - Prob. 17P Ch. 4.1 - Prob. 18P Ch. 4.1 - Prob. 19P Ch. 4.1 - Prob. 20P Ch. 4.1 - Prob. 21P Ch. 4.1 - Prob. 22P Ch. 4.1 - Prob. 23P Ch. 4.1 - Prob. 24P Ch. 4.1 - Prob. 25P Ch. 4.1 - Prob. 26P Ch. 4.1 - Prob. 27P Ch. 4.1 - Prob. 28P Ch. 4.1 - Prob. 29P Ch. 4.1 - Prob. 30P Ch. 4.1 - Prob. 31P Ch. 4.1 - Prob. 32P Ch. 4.1 - Prob. 33P Ch. 4.1 - Repeat Problem 33, except with the generator...Ch. 4.1 - A particle of mass m moves in the plane with...Ch. 4.1 - Prob. 36P Ch. 4.1 - Prob. 37P Ch. 4.2 - Prob. 1P Ch. 4.2 - Prob. 2P Ch. 4.2 - Prob. 3P Ch. 4.2 - Prob. 4P Ch. 4.2 - Prob. 5P Ch. 4.2 - Prob. 6P Ch. 4.2 - Prob. 7P Ch. 4.2 - Prob. 8P Ch. 4.2 - Prob. 9P Ch. 4.2 - Prob. 10P Ch. 4.2 - Prob. 11P Ch. 4.2 - Prob. 12P Ch. 4.2 - Prob. 13P Ch. 4.2 - Prob. 14P Ch. 4.2 - Prob. 15P Ch. 4.2 - Prob. 16P Ch. 4.2 - Prob. 17P Ch. 4.2 - Prob. 18P Ch. 4.2 - Prob. 19P Ch. 4.2 - Prob. 20P Ch. 4.2 - Suppose that L1=a1D2+b1D+c1 and L2=a2D2+b2D+c2,...Ch. 4.2 - Suppose that L1x=tDx+x and that L2x=Dx+tx. Show...Ch. 4.2 - Prob. 23P Ch. 4.2 - Prob. 24P Ch. 4.2 - Prob. 25P Ch. 4.2 - Prob. 26P Ch. 4.2 - Prob. 27P Ch. 4.2 - Prob. 28P Ch. 4.2 - Prob. 29P Ch. 4.2 - Prob. 30P Ch. 4.2 - Prob. 31P Ch. 4.2 - Prob. 32P Ch. 4.2 - Prob. 33P Ch. 4.2 - Prob. 34P Ch. 4.2 - Prob. 35P Ch. 4.2 - Prob. 36P Ch. 4.2 - Prob. 37P Ch. 4.2 - Prob. 38P Ch. 4.2 - Prob. 39P Ch. 4.2 - Prob. 40P Ch. 4.2 - Prob. 41P Ch. 4.2 - Prob. 42P Ch. 4.2 - Prob. 43P Ch. 4.2 - Prob. 44P Ch. 4.2 - Prob. 45P Ch. 4.2 - Prob. 46P Ch. 4.2 - Prob. 47P Ch. 4.2 - Prob. 48P Ch. 4.3 - Prob. 1P Ch. 4.3 - Prob. 2P Ch. 4.3 - Prob. 3P Ch. 4.3 - Prob. 4P Ch. 4.3 - Prob. 5P Ch. 4.3 - Prob. 6P Ch. 4.3 - Prob. 7P Ch. 4.3 - Prob. 8P Ch. 4.3 - Prob. 9P Ch. 4.3 - Prob. 10P Ch. 4.3 - Prob. 11P Ch. 4.3 - Prob. 12P Ch. 4.3 - Prob. 13P Ch. 4.3 - Prob. 14P Ch. 4.3 - Suppose that a projectile is fired straight upward...Ch. 4.3 - Prob. 16P Ch. 4.3 - Prob. 17P Ch. 4.3 - Prob. 18P Ch. 4.3 - Prob. 19P Ch. 4.3 - Prob. 20P Ch. 4.3 - Suppose that an artillery projectile is fired from...

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x ′ = y + z + e − t , y ′ = x + z , z ′ = x + y .

Solution Summary: The author explains how to obtain the general solution of the linear system of differential equation. Write the above equations in the form of D.

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x′=y+z+e−t,y′=x+z, z′=x+y .

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Chapter 4 Solutions