Program Explanation Given information: The linear system of the differential equation x ″ + 13 y ′ − 4 x = 6 sin t , y ″ − 2 x ′ − 9 y = 0 . Calculation: The given linear system of the differential equation x ″ + 13 y ′ − 4 x = 6 sin t , y ″ − 2 x ′ − 9 y = 0 . Write the system of differential equation in homogeneous equation. x ″ + 13 y ′ − 4 x − 6 sin t = 0 y ″ − 2 x ′ − 9 y = 0 The second differential equation of the given system is as follows, y ″ − 2 x ′ − 9 y = 0 ……. (1) Now write the above equations in the form of D as, D 2 x + 13 D y − 4 x = 6 sin t D 2 y − 2 D x − 9 y = 0 The above equation can be rewritten as, ( D 2 − 4 ) x + 13 D y = 6 sin t − 2 D x + ( D 2 − 9 ) y = 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 ) 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 ) can be solved for variable x in the form of determinant as, | D 2 − 4 13 D − 2 D D 2 − 9 | = | 6 sin t 13 D 0 D 2 − 9 | [ ( D 2 − 4 ) ( D 2 − 9 ) − 13 D ( − 2 D ) ] x = ( D 2 − 9 ) ( 6 sin t ) − 0 [ D 4 + 13 D 2 + 36 ] x = − 60 sin t Now the corresponding characteristic equation is as follows, r 4 + 13 r 2 + 36 = 0 ( r 2 + 4 ) ( r 2 + 9 ) = 0 Therefore, the roots of the equations are r 1 = − 2 i , r 2 = 2 i , r 3 = − 3 i and r 4 = 3 i . The complementary solution of the differential equation can be written as, x c ( t ) = a 1 cos 2 t + a 2 sin 2 t + b 1 cos 3 t + b 2 sin 3 t From equation (1), the value of f ( x ) = A sin t therefore assume the trial particular solution x p = A sin t . Since, the second derivative of x p = A sin t is x ″ = − A sin t and the fourth derivative is x ( 4 ) = A sin t Replace the value of x by x p ( t ) in the equation [ D 4 + 13 D 2 + 36 ] x = − 60 sin t as, [ D 4 + 13 D 2 + 36 ] ( A sin t ) = − 60 sin t A sin t − 13 A sin t + 36 A sin t = − 60 sin t 24 A sin t = − 60 sin t 24 A = − 60 Further simplify the above equation. A = − 60 24 A = − 5 2 Now substitute − 5 2 for A in the equation x p = A sin t to obtain the particular solution. x p = − 5 2 sin t The general solution of the differential equation is the sum of complementary function and the particular function. Therefore, the general solution of the differential equation is as follows, x ( t ) = a 1 cos 2 t + a 2 sin 2 t + b 1 cos 3 t + b 2 sin 3 t − 5 2 sin t Now apply the same method to find the differential equation. The system of equation { L 1 x + L 2 y = f 1 ( t ) L 3 x + L 4 y = f 2 ( t ) can be solved for variable y in the form of determinant as, | D 2 − 4 13 D − 2 D D 2 − 9 | = | D 2 − 4 6 sin t − 2 D D 2 − 9 | [ ( D 2 − 4 ) ( D 2 − 9 ) − 13 D ( − 2 D ) ] y = ( D 2 − 9 ) ( 6 sin t ) − 0 [ D 4 + 13 D 2 + 36 ] y = 12 cos t Now the corresponding characteristic equation is as follows, r 4 + 13 r 2 + 36 = 0 ( r 2 + 4 ) ( r 2 + 9 ) = 0 Therefore, the roots of the equations are r 1 = − 2 i , r 2 = 2 i , r 3 = − 3 i and r 4 = 3 i . The complementary solution of the differential equation can be written as, y c ( t ) = c 1 cos 2 t + c 2 sin 2 t + d 1 cos 3 t + d 2 sin 3 t Assume the trial particular solution y p = A cos t

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ISBN: 9780135988510

Author: Edwards

Publisher: PEARSON CO

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Problems on Growing Functions

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Chapter 4.2, Problem 16P

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x″+13y′−4x=6sint,y″−2x′−9y=0 .

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Chapter 4.2, Problem 15P

Chapter 4.2, Problem 17P

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Chapter 4 Solutions

EP DIFF.EQUAT.+BOUND.VALUE,...UPD.-ACC.

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 ″ + 13 y ′ − 4 x = 6 sin t , y ″ − 2 x ′ − 9 y = 0 .

Solution Summary: The author explains how to obtain the general solution of the linear system of differential equation.

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Program Description: Purpose of problem is to obtain the general solution of the linear system of the differential equation x″+13y′−4x=6sint,y″−2x′−9y=0 .

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Chapter 4 Solutions