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QUESTION 1 (20 MARKS) (a) Use z = y ¹ to solve the following initial value problem dy 4 dt (2) = -1. +y=²² (7 marks) (b) Find the general solution of the following first order homogeneous equation dy y(lnx - Iny). dz (6 marks) (e) Suppose that a body moves through a resisting medium with resistance pro- portional to its velocity v, so that and dv dt = -kv, where & is a positive constant. Given that the initial velocity and position of the body are to and zo respectively. Show that its velocity and position at time t are given by v(t) = te kt x(t)=x0+ (-) (1 - e-k). Hence find its finite distance as t→∞o. QUESTION 2 (15 MARKS) Suppose that an RLC circuit satisfies the basic circuit equation (7 marks) 1 LQ" + RQ+Q=E(t), dQ with R = 50 ohms, L= 0.1 henry, and C= 5 x 10- farad. Q(t) and I(t) are the charge and current at time t respectively where =I(t). At time t = 0, when both Q(0) and I(0) are both zero, the circuit is connected to a generator of E(t) 110 sinat. Find the current in the circuit at time t. dt (15 marks)