a. Explanation Given Information: Differential equation d 2 x d t 2 + 2 d x d t + 2 x = sin 2 t Initial condition is zero, x ( 0 ) = 2 x ′ ( 0 ) = − 3 Calculation: d 2 x d t 2 + 2 d x d t + 2 x = sin 2 t The given differential equation is linear and nonhomogeneous, so the general solution will be the solution of the homogeneous equation, d 2 x d t 2 + 2 d x d t + 2 x = 0 . d 2 x d t 2 + 2 d x d t + 2 x 8 x = 0 D 2 + 2 D + 2 = 0 ⇒ ( D + 1 ) 2 + 1 2 = 0 ⇒ x = c e − t cos t + d e − t sin t For a particular solution, x p = a cos 2 t + b sin 2 t ⇒ d x p d t = − 2 a sin 2 t + 2 b cos 2 t ⇒ d 2 x p d t 2 = − 4 a cos 2 t − 4 b sin 2 t ⇒ d 2 x p d t 2 + 2 d x p d t + 2 x p = − 4 a cos 2 t − 4 b sin 2 t + 2 ( − 2 a sin 2 t + 2 b cos 2 t ) + 2 ( a cos 2 t + b sin 2 t ) ⇒ d 2 x p d t 2 + 2 d x p d t + 2 x p = ( 4 b − 2 a b. To determine Solution of given differential equation. c. To determine Solution of given differential equation.

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ISBN: 9781118170519

Author: Norman S. Nise

Publisher: WILEY

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Solution of given differential equation.

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