Explanation Given: The undamped system is given as, d 2 y d t 2 + y = 5 cos t ; y ( 0 ) = 0 , y ′ ( 0 ) = 1 . Approach: The equation of undamped system is, m d 2 y d t 2 + k y = F 0 cos ( γ t ) … ( 1 ) Here, F 0 and γ are nonnegative constant and 0 < b 2 < 4 m k . The solution of the equation ( 1 ) will be of the form, y = y h + y p , where y p is the particular solution and y h is the general solution to the corresponding homogeneous equation. The general solution to the homogeneous equation m d 2 y d t 2 + k y = 0 is y h = A sin ( ω t + ϕ ) , where ω = k m and the particular solution is y p = F 0 2 m ω t sin ω t . Calculation: Consider the equation, d 2 y d t 2 + y = 5 cos t ; ... ( 2 ) Compare the equation ( 2 ) with equation ( 1 ) . m = 1 , k = 1 , F 0 = 5 , γ = 1 , and ω = k m = 1 1 = 1 Substitute 1 for ω in equation y h = A sin ( ω t + ϕ ) . y h = A sin ( t + ϕ ) … ( 3 ) Differentiate the equation ( 3 ) with respect to t . y h ′ = A cos ( t + ϕ ) Again differentiate, y h ′ ′ = − A sin ( t + ϕ ) Substitute y h and y h ′ ′ in place of y and y ″ in equation ( 2 )

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

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Chapter 4.10, Problem 4E

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Fundamentals of Differential Equations [With CDROM] - 7th Edition

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The equation of the motion for a given undamped system and sketch the solution.

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