1. Use the definition of the Laplace transform to find the Laplace transform of the given function. (a) f(t) = = t tЄ [0, 1) 3-t te [1,2) 1 tЄ [2,3) 0 t = [3,∞) t 0<t<1 (b) g(t) == 1 (c) f(t) = So t > 1 0 tЄ [0, 1) -t (et t€ [1,∞) 2. Use the Laplace transform to solve the given initial value problem. (a) y" - 4y' + 4y = 0, y(0) = 1, y'(0) = 1 (b) y" + 2y + y = 4e, y(0) = 2, y'(0) = -1 (c) y" +4y+8y = 17e sin 2t, y(0) = 2, = 2, y'(0) = -1 (d) y" w2y= cos 2t, w² 4, y(0) = 1, y'(0) = 0 3. Use the Laplace transform to solve the given inital value problem. 0te [0,7) = 1 tЄ [T, 2π), y(0) = 0, y'(0) = 1 (a) y" +2y+2y= 0 t = [2π, ∞) (b) y" + 3y+2y=U(t-2), y(0) = 1, y'(0) = = 5 (c) Use some form of software to plot y(t) from part (b) for tЄ [0, 15].