Program Explanation Given information: The linear system of the differential equation x ″ = − 4 x + sin t , y ″ = 4 x − 8 y . Calculation: The given linear system of the differential equation x ″ = − 4 x + sin t , y ″ = 4 x − 8 y . Write the system of differential equation in homogeneous equation. x ″ + 4 x − sin t = 0 y ″ − 4 x + 8 y = 0 The first differential equation of the given system is as follows, x ″ + 4 x − sin t = 0 ...............(1) Now write the above equations in the form of D as, D 2 x + 4 x = sin t − 4 x + D 2 y + 8 y = 0 Now write the above equation in terms of x and y as, ( D 2 + 4 ) x + 0 ⋅ y = sin t − 4 x + ( D 2 + 8 ) y = 0 Label the first equation of the above system as, ( D 2 + 4 ) x = sin t ............(2) Label the second equation of the above system as, − 4 x + D 2 y + 8 y = 0 …… (3) Now the corresponding characteristic equation of x ( t ) from first equation is as follows, ( r 2 + 4 ) = 0 Now find the roots of the characteristic equation is as follows, ( r 2 + 4 ) = 0 Therefore, the roots of the equations are r 1 = − 2 i , r 2 = 2 i . The complementary function of x ( t ) can be written as, x c ( t ) = c 1 cos 2 t + c 2 sin 2 t Consider the trial particular function of the differential equation x ″ + 4 x − sin t = 0 as x p ( t ) = A cos t + B sin t Now replace the value of x p ( t ) = A cos t + B sin t with x ( t ) in equation (1). ( D 2 + 4 ) x p = sin t ( D 2 + 4 ) ( A cos t + B sin t ) = sin t ( − A cos t − B sin t ) + 4 ( A cos t + B sin t ) = sin t − 3 A cos t + 3 B sin t = sin t Now compare the corresponding coefficients and then solve the equation as, − 3 A = 0 , B = 1 3 A = 0 , B = 1 3 Therefore, the particular solution is x p ( t ) = 1 3 sin t . The general solution is the sum of complementary function and particular solution. The general solution can be written as, x ( t ) = c 1 cos 2 t + c 2 sin 2 t + 1 3 sin t Now multiply ( D 2 + 4 ) with equation (3) to eliminate x

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 4.2, Problem 14P

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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+sint,y″=4x−8y .

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Chapter 4.2, Problem 13P

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Chapter 4 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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 + sin t , y ″ = 4 x − 8 y .

Solution Summary: The author analyzes the linear system of the differential equation x′′=-4x+mathrmsin

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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+sint,y″=4x−8y .

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Chapter 4 Solutions