) Use the Laplace transform to solve the following initial value problem: Use step(t-c) for ue(t). y(t) = y" +10y +34y= S(t - 7) y(0) = 0, y'(0) = 0

Use the Laplace transform to solve the given equation. y'(t) = cos t + y(t) cos(t - t)dt, y(0) = 1 y(t) =

8. Find the Laplace transform of the function g(t) = e"[4t + 5 cos(3t) + 6 sin(3t)] %3|

Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function -4-5(t-1), (0)-6, (0) 0. = a Find the Laplace transform of the solution Y()-C (v(t)}- b. Obtain the solution y(t). (1) c Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at 1. if 0 < t < 1, -{₁ u(t)= if 1≤ t < 00.