Program Explanation Given information: The differential function and initial conditions are, x ″ + 8 x ′ + 25 x = 200 cos t + 520 sin t ; x ( 0 ) = − 30 , x ′ ( 0 ) = − 10 Explanation: The given differential function and initial conditions are, x ″ + 8 x ′ + 25 x = 200 cos t + 520 sin t ; x ( 0 ) = − 30 , x ′ ( 0 ) = − 10 The corresponding auxiliary equation is, r 2 + 8 r + 25 = 0 The roots of the auxiliary equation is calculated as, r = − b ± b 2 − 4 a c 2 a r = − 8 ± ( 8 ) 2 − 4 ( 1 ) ( 25 ) 2 ( 1 ) r = − 8 ± − 36 2 r = − 4 ± 3 i Therefore, the roots of the auxiliary equation are − 4 − 3 i and − 4 + 3 i . Hence, the complementary function is, x c ( t ) = e − 4 t ( c 1 cos 3 t + c 2 sin 3 t ) Consider the particular function as x p = A cos t + B sin t . Differentiate with respect to t as, x ′ p = − A sin t + B cos t Again, differentiate with respect to t as, x ″ p = − A cos t − B sin t Substitute the values of x ″ p and x p in the given equation x ″ + 8 x ′ + 25 x = 200 cos t + 520 sin t as, ( − A cos t − B sin t ) + 8 ( − A sin t + B cos t ) + 25 ( A cos t + B sin t ) = 200 cos t + 520 sin t ( 24 A + 8 B ) cos t + ( 24 B − 8 A ) sin t = 200 cos t + 520 sin t Compare the corresponding coefficients on each side of the above equation as, 24 A + 8 B = 0 and − 8 A + 24 B = 10 A = 1 and B = 22 Now, substitute the values of A and B in the equation x p = A cos t + B sin t . x p = cos t + 22 sin t Therefore, the steady periodic oscillation of the mass is, x s p ( t ) = cos t + 22 sin t The general solution of the given differential equation is, x ( t ) = c 1 cos 3 t e − 4 t + c 2 sin 3 t e − 4 t + cos t + 22 sin t Differentiate with respect to t as, ` x ′ ( t ) = c 1 ( − 4 e − 4 t cos 3 t − 3 e − 4 t sin 3 t ) + c 2 ( − 4 e − 4 t sin 4 t + 3 e − 4 t cos 4 t ) + 22 cos t − sin t Substitute the initial conditions x ( 0 ) = 30 and x ′ ( 0 ) = − 10

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 3.6, Problem 14P

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Program Description: Purpose of problem is to find and plot the steady periodic solution and the actual solution that satisfies the differential function x″+8x′+25x=200cost+520sint .

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Chapter 3.6, Problem 13P

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Chapter 3 Solutions

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

Ch. 3.1 - In Problems 1 through 16, a homogeneous...Ch. 3.1 - Prob. 2P Ch. 3.1 - Prob. 3P Ch. 3.1 - Prob. 4P Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 7P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P

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Program Description: Purpose of problem is to find and plot the steady periodic solution and the actual solution that satisfies the differential function x ″ + 8 x ′ + 25 x = 200 cos t + 520 sin t .

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Program Description: Purpose of problem is to find and plot the steady periodic solution and the actual solution that satisfies the differential function x″+8x′+25x=200cost+520sint .

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Chapter 3 Solutions