171_pdfsam_math 54 differential equation solutions odd

171_pdfsam_math 54 differential equation solutions odd - 2...

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CHAPTER 4: Linear Second Order Equations EXERCISES 4.1: Introduction: The Mass-Spring Oscillator, page 159 1. With b =0and F ext = 0, equation (3) on page 155 becomes my 0 + ky =0 . Substitution y =s in ωt ,where ω = p k/m , yields m (sin ωt ) 0 + k (sin ωt )= 2 sin ωt + k sin ωt =s i n ωt ( 2 + k ) =s in ωt ( m ( k/m )+ k )=0 . Thus y =s in ωt is indeed a solution. 3. Diferentiating y ( t ), we Fnd y =2s in3 t +cos3 t y 0 =6cos3 t 3sin3 t y 0 = 18 sin 3 t 9cos3 t. Substituting y , y 0
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Unformatted text preview: 2 y + 18 y = 2( 18 sin 3 t 9 cos 3 t ) + 18(2 sin 3 t + cos 3 t ) = [2( 18) + 18(2)] sin 3 t + [2( 9) + 18(1)] cos 3 t = 0 . Next, we check that the initial conditions are satisFed. y (0) = (2 sin 3 t + cos 3 t ) t =0 = 2 sin 0 + cos 0 = 1 , y (0) = (6 cos 3 t 3 sin 3 t ) t =0 = 6 cos 0 3 sin 0 = 6 . 167...
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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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