y t 2 δ t 3 y 0 2 y 0 6 c y 00 2 y 5 y 5 t 4 t 4 y 0 0 y 0 4 d y 00 4 y cos t y

# Y t 2 δ t 3 y 0 2 y 0 6 c y 00 2 y 5 y 5 t 4 t 4 y 0

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y = t 2 δ ( t - 3) , y (0) = 2 , y 0 (0) = - 6 c. y 00 + 2 y 0 + 5 y = 5 , 0 t < 4 0 , t 4 y (0) = 0 , y 0 (0) = 4 d. y 00 + 4 y = cos( t ) , y (0) = 1 , y 0 (0) = 0 e. y 00 + 2 y 0 - 3 y = t 2 , 0 t < 3 0 , t 3 y (0) = 0 , y 0 (0) = 0

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5. a. Show that, if L{ f } = F ( s ), then L{ t f ( t ) } = - dF ( s ) ds . b. From the result above, derive the formula for L{ t sin(2 t ) } . c. Use this result to solve the initial value problem y 00 + 4 y = 2 cos(2 t ) , y (0) = - 1 , y 0 (0) = 4 . 6. A bridge can be considered to be a harmonic oscillator. When someone is walking across the bridge, their steps impart an impulsive force. Below we examine two cases of impulsive force applied to the harmonic oscillator. a. Solve the initial value problem: d 2 y dt 2 + y = X j =0 δ ( t - ) , y (0) = 0 y 0 (0) = 0 . b. Solve this initial value problem: d 2 y dt 2 + y = X j =0 δ ( t - 2 ) , y (0) = 0 y 0 (0) = 0 . c. Use the information above to explain why soldiers are instructed to break cadence when marching across a bridge. (A historical note: In the 17 th century, a number of British soldiers died when they marched across a bridge and set up resonance so that the bridge collapsed.)
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