Program Explanation Given information: The linear system of the differential equation x ′ = 4 x − 2 y , y ′ = − 4 x + 4 y − 2 z , z ′ = − 4 y + 4 z . Calculation: The given linear system of the differential equation x ′ = 4 x − 2 y , y ′ = − 4 x + 4 y − 2 z , z ′ = − 4 y + 4 z . Write the system of differential equation in homogeneous equation. x ′ − 4 x + 2 y = 0 y ′ + 4 x − 4 y + 2 z = 0 z ′ + 4 y − 4 z = 0 Now write the above equations in the form of D as, ( D − 4 ) x + 2 y = 0 4 x + ( D − 4 ) y + 2 z = 0 4 y + ( D − 4 ) z = 0 The resultant equation represents in the form. L 1 x + L 2 y = f 1 ( t ) L 3 x + L 4 y = f 2 ( t ) L 5 x + L 6 y = 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 1 x + L 2 y = f 1 ( t ) L 3 x + L 4 y = f 2 ( t ) L 5 x + L 6 y = f 3 ( t ) can be written in the form of determinant as, | D − 4 2 0 4 D − 4 2 0 4 D − 4 | x = 0 Now the corresponding characteristic equation is as follows, ( r − 4 ) 3 − 16 ( r − 4 ) = 0 ( r − 4 ) [ ( r − 4 ) 2 − 16 ] = 0 ( r − 4 ) [ ( r − 4 ) 2 − 4 2 ] = 0 ( r − 4 ) ( r − 8 ) = 0 Therefore, the roots of the equations are r 1 = 0 , r 2 = 4 , r 3 = 8 . The solution of the differential equation can be written as, x ( t ) = c 1 + c 2 e 4 t + c 3 e 8 t y ( t ) = d 1 + d 2 e 4 t + d 3 e 8 t z ( t ) = e 1 + e 2 e 4 t + e 3 e 8 t Since the degree of the operational determinant is 3. Therefore, the given system has 3 arbitrary constant. Differentiate x ( t ) = c 1 + c 2 e 4 t + c 3 e 8 t with respect to t both sides. x ′ ( t ) = 4 c 2 e 4 t + 8 c 3 e 8 t Differentiate y ( t ) = d 1 + d 2 e 4 t + d 3 e 8 t with respect to t both sides. y ′ ( t ) = 4 d 2 e 4 t + 8 d 3 e 8 t Differentiate z ( t ) = e 1 + e 2 e 4 t + e 3 e 8 t with respect to t both sides

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ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 4.2, Problem 19P

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x′=4x−2y,y′=−4x+4y−2z, z′=−4y+4z .

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Chapter 4.2, Problem 18P

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Chapter 4 Solutions

EBK DIFFERENTIAL EQUATIONS

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 ′ = 4 x − 2 y , y ′ = − 4 x + 4 y − 2 z , z ′ = − 4 y + 4 z .

Solution Summary: The author explains how the linear system of the differential equation can be obtained by the determinant method.

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x′=4x−2y,y′=−4x+4y−2z, z′=−4y+4z .

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Chapter 4 Solutions