Assignment8-solution-public

# Assignment8-solution-public - Math 201 Assignment#8...

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Unformatted text preview: Math 201 Assignment #8 Solutions December 9, 2011 Grade Problems 1 and 3, • 10 marks each, with additional 3 effort each marks for problems 2, 4, and 5(a). Total 29. • 5(b) and 5(c) are not required to finish. 1. Solve the integral differential equation- ty + ´ t y ( τ ) e t- τ dτ = e t- 1 , y (0) = 0 . • Take Laplace transform on both sides, d ds ( sY ) + Y 1 s- 1 = 1 s- 1- 1 s = 1 s ( s- 1) Simplify, Y + Y 1 s- 1 = 1 s 2 ( s- 1) • Solve this first order DE, [( s- 1) Y ] = 1 s 2 Y =- 1 s ( s- 1) + C s- 1 where C is an arbitrary constant. y = L- 1 {- 1 s ( s- 1) } + L- 1 { C s- 1 } = L- 1 { 1 s- 1 s- 1 } + Ce t = 1- e t + Ce t = 1 + c 1 e t where c 1 is an arbitrary constant. 2. A mass of 1kg is suspended under a spring with a constant 1N/m. Assume that there is no damping force. The spring is at rest initially. The mass is knocked downward by a hammer with a force of 1N twice, at t = 0 and t = π , respectively. Determine the motion of the mass. • Let x ( t ) be the motion of the mass, with positive direction pointing downwards, x 00 + x = δ ( t ) + δ ( t + π ) with initial conditions x (0) = x (0) = 0 . 1 • Take Laplace on both sides, s 2 X + X = 1 + e- π • Isolate X , X = 1 s 2 + 1 + e- π 1 s 2 + 1 • Take inverse Laplace transform, x = sin t + sin( t- π ) U ( t- π ) = sin t- sin t U ( t- π ) = sin t [1- U ( t- π )] • Note that this function is equivalent to sin t for ≤ t < π and otherwise....
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Assignment8-solution-public - Math 201 Assignment#8...

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