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s11week12 - Math 205 Spring 2011 Homework 12 due April 18...

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Math 205, Spring 2011 Homework 12: due April 18 Chapter 6, Sections 5, 6 and 7.
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2 Week 12: 6.5 Springs 6.6 Circuits 6.7 Variation of Parameters
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3 The general homogeneous mass-spring system is m d 2 y dt 2 + c dy dt + ky = 0 , where m is the mass, k is the spring constant and c is the damping constant (also called “friction”). In the case c = 0 the system is said to be undamped. When c negationslash = 0 the system is damped, and the motion depends primarily on the type of damping, determined by the roots. Problem 5 Solve y ′′ + 2 y + 5 y = 0 , with IV y (0) = 1 ,y (0) = 3 , and determine whether the system is overdamped, critically damped or damped and oscillating, describe the sketch. (soln.) char. polyn. r 2 + 2 r + 5 = 0 , roots are 1 ± 2 i, complex, so the system is underdamped. With y = e t ( c 1 cos2 t + c 2 sin2 t ) we get 1 = y (0) = c 1 , y = e t (cos2 t + c 2 sin2 t ) , so y = e t (cos2 t + c 2 sin2 t ) + e t ( 2sin2 t + 2 c 2 cos2 t )) , (continued ... ) 3 = y (0) = (1 + 0) + (0 + 2 c 2 ) , so c 2 = 2 , y = e t (cos2 t + 2sin2 t ) .
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