Explanation The given differential equation is u ″ + ω 0 2 u = cos ω 0 t , . Consider the homogeneous equation u ″ + ω 0 2 u = 0 . Obtain the characteristic equation by replacing y ″ with r 2 , y ′ with r and y with 1 in the differential equation: r 2 + ω 0 2 = 0 . The roots of the characteristic equation are as follows. r 2 + ω 0 2 = 0 r 1 , 2 = 0 ± ( 0 ) 2 − 4 ( 1 ) ( ω 0 2 ) 2 ( 1 ) ( ∵ x 1 , 2 = − b ± b 2 − 4 a c 2 a ) r 1 , 2 = ± 0 − 4 ω 0 2 2 r 1 , 2 = ± − 4 ω 0 2 2 Simplify further, r 1 , 2 = ± i 2 ω 0 2 2 r 1 , 2 = ± ω 0 i ( ∵ r = a + b i ) Here, roots are complex. Note that, the general equation is y c ( t ) = c 1 e a t cos b t + c 2 e a t sin b t where a is the real part of the root and b is the imaginary part of the roots. Substitute a = 0 and b = ω 0 in y c ( t ) = c 1 e a t cos b t + c 2 e a t sin b t . y c ( t ) = c 1 e 0 t cos ( ω 0 t ) + c 2 e 0 t sin ( ω 0 t ) = c 1 cos ( ω 0 t ) + c 2 sin ( ω 0 t ) Therefore, the complementary solution of the given differential equation is y c ( t ) = c 1 cos ( ω 0 t ) + c 2 sin ( ω 0 t ) . Let U = A t cos ( ω 0 t ) + B t sin ( ω 0 t ) ( ∵ g ( t ) = cos ω 0 t ) . Obtain the first derivative of U = A t cos ( ω 0 t ) + B t sin ( ω 0 t ) as follows. U ′ = d d t ( A t cos ( ω 0 t ) + B t sin ( ω 0 t ) ) = A [ t ( − ω 0 sin ω 0 t ) + cos ( ω 0 t ) ( 1 ) ] + B [ t ( ω 0 cos ω 0 t ) + sin ( ω 0 t ) ( 1 ) ] = − A t ω 0 sin ω 0 t + A cos ω 0 t + B t ω 0 cos ω 0 t + B sin ( ω 0 t ) Obtain the second derivative of U = A t cos ( ω 0 t ) + B t sin ( ω 0 t ) as follows. U ″ = d d t ( U ′ ) = d d t ( − A t ω 0 sin ω 0 t + A cos ω 0 t + B t ω 0 cos ω 0 t + B sin ( ω 0 t ) ) = d d t ( − A t ω 0 sin ω 0 t ) + d d t ( A cos ω 0 t ) + d d t ( B t ω 0 cos ω 0 t ) + d d t ( B sin ( ω 0 t ) ) = − ω 0 A [ t ( ω 0 cos ω 0 t ) + sin ( ω 0 t ) ( 1 ) ] − A ω 0 sin ω 0 t + B ω 0 [ t ( − ω 0 sin ω 0 t ) + cos ( ω 0 t ) ( 1 ) ] + B ( ω 0 cos ω 0 t ) = ( − ω 0 2 A t cos ω 0 t − ω 0 A sin ( ω 0 t ) − A ω 0 sin ω 0 t − B t ω 0 2 sin ω 0 t + B ω 0 cos ( ω 0 t ) + B ( ω 0 cos ω 0 t ) ) Obtain the particular solution as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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