Write each function in terms of unit step functions. (a) f(t) = 2, 0 <= t < 3, -2, t >= 3. (b) g(t) = 0, 0 <= t < 3pi/2, sin(t), t >= 3pi/2. (c) h(t) = 1, 0 <= t < 4, 0 4 <= t < 5, 1, t >= 5.
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.