236_pdfsam_math 54 differential equation solutions odd

236_pdfsam_math 54 differential equation solutions odd - 1....

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Chapter 4 and, in the right-hand side, lim t +0 r 2 GM a t + C 2 = r 2 GM a · 0+ C 2 = C 2 . Thus C 2 = aπ/ 2and r ( t )sat isfes a arctan s r ( t ) a r ( t ) p r ( t )[ a r ( t )] a = r 2 GM a t + 2 . At the moment t = T 0 , when Earth splashes into the sun, we have r ( T 0 ) = 0. Substituting this condition into the last equation yields a arctan r 0 a 0 p 0( a 0) a = r 2 GM a T 0 + 2 0= r 2 GM a T 0 + 2 T 0 = 2 r a 2 GM = π 2 2 r a 3 GM . Then the required ratio is T 0 T = π 2 2 r a 3 GM . 2 π r a 3 GM = 1 4 2 . EXERCISES 4.8:
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Unformatted text preview: 1. In this problem, we have undamped Free vibration case governed by equation (2) on page 210 in the text. With m = 3 and k = 48, the equation becomes 3 y + 48 y = 0 (4.10) with the initial conditions y (0) = . 5, y (0) = 2. The angular velocity oF the motion is = r k m = r 48 3 = 4 . 232...
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