Solve the following initial-value problems using Laplace transforms. (

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Solve the following initial-value problems using Laplace transforms. (a) y'' + y = g(t), y(0) = y'(0) = 0. Here g(t) = t/2, 0 <= t < 6, 3, t >= 6. (b) y'' + 2y = u(t - pi) - u(t - 2pi), y(0) = 0, y'(0) = 0. (c) y'' + 4y' + 3y = u(t - pi)e ^-2(t - pi) , y(0) = 0, y'(0) = 0. (d) y'' + 3y = del(t - pi), y(0) = y'(0) = 0. (e) y'' + 2y = u(t - pi) + 3del(t - 3pi/2) - u(t - 2pi), y(0) = y'(0) = 0.

Solve the following initial-value problems using Laplace transforms. (a) y'' + y = g(t), y(0) = y'(0) = 0. Here g(t) = t/2, 0 = 6. (b) y'' + 2y = u(t - pi) - u(t - 2pi), y(0) = 0, y'(0) = 0. (c) y'' + 4y' + 3y = u(t - pi)e ^-2(t - pi) , y(0) = 0, y'(0) = 0. (d) y'' + 3y = del(t - pi), y(0) = y'(0) = 0. (e) y'' + 2y = u(t - pi) + 3del(t - 3pi/2) - u(t - 2pi), y(0) = y'(0) = 0.