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243_pdfsam_math 54 differential equation solutions odd

243_pdfsam_math 54 differential equation solutions odd - y...

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Exercises 4.8 Therefore, the equation of motion is given by y ( t ) = 1 99 . 99 e 0 . 1 t sin 99 . 99 t . The maximum displacement to the right occurs at the first point of local maximum of y ( t ). The critical points of y ( t ) are solutions to y ( t ) = e 0 . 1 t 99 . 99 99 . 99 cos 99 . 99 t 0 . 1 sin 99 . 99 t = 0 99 . 99 cos 99 . 99 t 0 . 1 sin 99 . 99 t = 0 tan 99 . 99 t = 10 99 . 99 = 9999 . Solving for t , we conclude that the first point of local maximum is at t = (1 / 99 . 99) arctan 9999 0 . 156 sec . 13. In Example 3, the solution was found to be y ( t ) = 7 12 e 2 t sin 2 3 t + φ , (4.13) where φ = π + arctan( 3 / 2). Therefore, we have y ( t ) = 7 3 e 2 t sin 2 3 t + φ + 7 e 2 t cos 2 3 t
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Unformatted text preview: y ( t ), we set y ( t ) = − r 7 3 e − 2 t sin ± 2 √ 3 t + φ ² + √ 7 e − 2 t cos ± 2 √ 3 t + φ ² = 0 ⇒ sin ( 2 √ 3 t + φ ) cos ( 2 √ 3 t + φ ) = √ 7 p 7 / 3 = √ 3 ⇒ tan ± 2 √ 3 t + φ ² = √ 3 . Since tan θ = √ 3 when θ = ( π/ 3)+ nπ , where n is an integer, we see that the relative extrema will occur at the points t n , where 2 √ 3 t n + φ = π 3 + nπ ⇒ t n = ( π/ 3) + nπ − φ 2 √ 3 . 239...
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