a) Derive the Laplace transform of the following functions of time from first principles, explaining your analysis fully: (i) f (t ) = 1-3e-21 (5 marks) (ii) lim Unit impulse function: S(t) = h->0 , Ostsh 0, otherwise (5 marks) b) Using the attached Laplace transform table, determine the Laplace transform of the following 2nd order differential equation, assuming zero initial conditions: d' x dx - +5 + 4x = sint dt2 dt Hence derive an expression for X(s). (7 marks) c) Using partial fractions and the Laplace transform table, as appropriate, derive the solution, x, of the above differential equation. (8 marks)