212_pdfsam_math 54 differential equation solutions odd

212_pdfsam_math 54 differential equation solutions odd - V...

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Chapter 4 Thus, a general solution to the equation describing the motion is y ( t )= cos t + 1 2 sin t + c 1 e 3 t + c 2 e t . Diferentiating, we Fnd y 0 ( t )=s in t +(1 / 2) cos t 3 c 1 e 3 t c 2 e t . Initial conditions give y (0) = 1+ c 1 + c 2 =1 / 2 , y 0 (0) = 1 / 2 3 c 1 c 2 =0 c 1 = 1 / 2 , c 2 =2 . Hence, the equation o± motion is y ( t )= cos t + 1 2 sin t 1 2 e 3 t +2 e t . 45. (a) With m = k =1and L = π given initial value problem becomes y ( t )=0 ,t ≤− π 2 V , y 0 + y 0 = ( cos Vt, π/ (2 V ) <t<π/ (2 V ) , 0 ,t π/ (2 V ) . The corresponding homogeneous equation y 0 + y =0i sthes imp leha rmon icequa t ion whose general solution is y h ( t )= C 1 cos t + C 2 sin
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Unformatted text preview: V ) < t < / (2 V ). The nonhomogeneous term, cos V t , suggests a particular solution o the orm y p ( t ) = A cos V t + B sin V t. Substituting y p ( t ) into the equation yields ( A cos V t + B sin V t ) + ( A cos V t + B sin V t ) = cos V t V 2 A cos V t V 2 B sin V t ) + ( A cos V t + B sin V t ) = cos V t 1 V 2 ) A cos V t + 1 V 2 ) B sin V t = cos V t . Equating coecients, we get A = 1 1 V 2 , B = 0 , 208...
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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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