(d) The 4 kg mass is released from rest at a distance xo to the right of the equilibrium position. c = 40Nsm-1 4 kg k = 90Nm-1 (I) Set up a differential equation to represent the dynamics of the system above. (1 mark) (II) Solve the differential equation in (I) above to obtain the displacement, x, as a function of time. (2 marks) (e) Given the circuit R - C circuit in the figure below: (f) (g) ww E R (i) In terms of , R and C, derive the differential equation for the circuit. (11) (1 mark) Solve the differential equation in (i) above for the current, I, flowing through the circuit after time t, given that at time t = 0, Q = 0 using the Laplace transform. (111) (iv) (1 mark) (2 marks) Using the solution in (ii) above, determine the value of the current at t = 0.2s given that R = 29, C = 10F and = 12V. Replace the resistor in the circuit with an inductor of inductance L and remove the source & so that the capacitor becomes the source. If at time t = 0, Q = Qo and I = 0, show using Laplace transforms that I(t) = sin sinc (5 marks) (11) List down the two recurrence formulae for Bessel functions, Jn(x) (1 mark) Hence prove the identity: 2Jn' = Jn-1 - Jn+1 Evaluate 1.5 dx using Simpson's rule with 7 ordinates e-1 (1 mark) (2 marks) 0