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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 = 0 and F ext = 0, equation (3) on page 155 becomes my + ky = 0 . Substitution y = sin ωt , where ω = k/m , yields m (sin ωt ) + k (sin ωt ) = 2 sin ωt + k sin ωt = sin ωt ( 2 + k ) = sin ωt ( m ( k/m ) + k ) = 0 . Thus y = sin ωt is indeed a solution. 3. Differentiating y ( t ), we find y = 2 sin 3 t + cos 3 t y = 6 cos 3 t 3 sin 3 t y = 18 sin 3 t 9 cos 3 t. Substituting y , y , and y into the given equation, we get
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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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