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319_pdfsam_math 54 differential equation solutions odd

# 319_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.5 By solving these equations simultaneously, we find c 1 = 1 , c 2 = 1 , c 3 = 1 , and c 4 = 0 . Therefore, the solution to this spring-mass-dashpot system is x ( t ) = e t te t cos t, y ( t ) = e t + te t cos t. 7. In operator notations, ( D 2 + 5 ) [ x ] 2 y = 0 , 2 x + ( D 2 + 2 ) [ y ] = 3 sin 2 t. Multiplying the first equation by ( D 2 + 2) and the second equation by 2, and adding the results, we obtain ( D 2 + 2 ) ( D 2 + 5 ) 4 [ x ] = 6 sin 2 t ( D 4 + 7 D 2 + 6 ) [ x ] = 6 sin 2 t ( D 2 + 1 ) ( D 2 + 6 ) [ x ] = 6 sin 2 t . (5.35) Since the characteristic equation, ( r 2 + 1)( r 2 + 6) = 0, has the roots r = ± i and r = ± i 6, a general solution to the corresponding homogeneous equation is given by x h ( t ) =
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