(a) Calculate the inverse Laplace transforms of the following functions. Explain your reasoning. (i) 1-28 482 +36 2s (ii) (iii) (s+3)(s1)2 s+1 82-68+5 (iv) se-48 82-36 [14 marks] (b) Use the Laplace transform to solve the following differential equation. dx dx dx dt dt +4- +8x(t)H(t 1) where x(0) = 0 and (0) = 0. dt Display your calculations clearly, explain any abbreviations used, and name the appropriate theorems or properties as you use them. [16 marks]