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 Frst point of local maximum of y ( t ). The critical points of y ( t ) are solutions to y 0 ( t )= e 0 . 1 t 99 . 99 ± 99 . 99 cos 99 . 99 t 0 . 1sin 99 . 99 t ² =0 99 . 99 cos 99 . 99 t 0 . 1sin 99 . 99 t =0 tan 99 . 99 t =10 99 . 99 = 9999 . Solving for t , we conclude that the Frst 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 )= r 7 12 e 2 t sin ± 2 3 t + φ ² , (4.13) where φ = π + arctan( 3 / 2). Therefore, we have y 0 ( t )= r 7 3 e 2 t sin ± 2 3 t + φ ² + 7 e 2 t cos ± 2 3 t + φ ² . Thus, to Fnd the relative extrema for
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Unformatted text preview: y ( t ), we set y ( t ) = r 7 3 e 2 t sin 2 3 t + + 7 e 2 t cos 2 3 t + = 0 sin ( 2 3 t + ) cos ( 2 3 t + ) = 7 p 7 / 3 = 3 tan 2 3 t + = 3 . Since tan = 3 when = ( / 3)+ n , where n is an integer, we see that the relative extrema will occur at the points t n , where 2 3 t n + = 3 + n t n = ( / 3) + n 2 3 . 239...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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