240_pdfsam_math 54 differential equation solutions odd

240_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 k =30 k =30 k =30 . Inth iscase , r = 5 ± 25 30 = 5 ± 5 i . A general solution has the form y =( C 1 cos 5 t + C 2 sin 5 t ) e 5 t .F o rcon s tan t s C 1 and C 2 , we have the system y (0) = ( C 1 cos 5 t + C 2 sin 5 t ) e 5 t ± ± t =0 = C 1 =1 , y 0 (0) = ² ( 5 C 2 5 C 1 )cos 5 t ( 5 C 1 +5 C 2 )sin 5 t ³ e 5 t ± ± t =0 = 5 C 2 5 C 1 =0 C 1 =1 , C 2 = 5 , and so y ( t )= h cos 5 t + 5sin 5 t i e 5 t = 6 e 5 t sin ´ 5 t + φ µ , where φ = arctan(1 / 5) 0 . 421 . Graphs of the solutions for k = 20, 25, and 30 are shown in Figures B.23–B.25 in the answers in the text. 7. The motion of this mass-spring system is governed by equation (12) on page 213 in the text. With m =1 / 8, b =2 ,and k = 16 this equation becomes 1 8 y 0 +2 y 0 +16 y =0 , (4.12) and the initial conditions are y (0) = 3 / 4, y 0 (0) = 2. Since
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