solf4 - EXAM FOUR SAMPLE SOLUTIONS 1 Use convolution to...

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Unformatted text preview: EXAM FOUR SAMPLE SOLUTIONS 1. Use convolution to express a particular solution to x 00 + x = sec ( t ) as an integral–then evaluate. Let F ( s ) = L{ sec t } . Then s 2 X + X = F ( s ), so that X = F ( s ) / ( s 2 +1) = F ( s ) G ( s ) where g ( t ) = sin t . Thus x ( t ) = sec t * sin t = R t sec usin ( t- u ) du = R t sec u ( sin tcos u- cos tsin u ) du = sin t R t du- cos t R t tan udu = tsin t- cos tln ( sec t ). 2. Solve using Laplace transforms: y 00 + 25 y = 10 δ ( t- 2); y (0) = y (0) = 0 s 2 Y + 25 Y = 10 e- 2 s , so Y = 10 e- 2 s / ( s 2 + 25) = e- 2 s F ( s ). Then f ( t ) = 2 sin 5 t and y ( t ) = u ( t- 2) f ( t- 2) = u ( t- 2) sin 5( t- 2). That is, f ( t ) = 0 for t < 2 and f ( t ) = sin 5( t- 2) for t > 2 3. A mass of 4 grams on a spring with constant k = 100 is released from rest at time t = 0, 2 cm above equilibrium. Then at time t = 3, the mass is given an upward impulse of power 120. Write the differential equation for the position x ( t ) of the mass at time t and use Laplace transforms to solve for x ( t )....
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solf4 - EXAM FOUR SAMPLE SOLUTIONS 1 Use convolution to...

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