QUESTION 7: 11 marks Consider the differential equation (D, ) : 4x(t) +6x(t) -32 vx(t) =0, x(0) = 4 and the relation y(t) = 2vx(t) + 5. 1. Show that y(t) satisfies the differential equation (D2) : 4y(t)+3y(t)-47 = 0, y(0) = 9 if and only if r(t) satisfies the differential equation (D1). 2 marks 2. Find the backward solution of (D2) and deduce the backward solution of (D1). 3 marks 3. Is the solution of (D1) convergent or divergent? Justify your answer. 2 marks 4. Determine the stationary solutions of (D1 ) and indicate whether they are stable or unstable. 2 marks 5. Sketch a phase diagram and a time-path diagram of (D1). 2 marks For questions 4 and 5, you can exploit Figure 1 below. If(x) 15 10 (C) 5 5 10 15 20 25 35 40 45 -5 -10 -15 Figure 1: The graph (C) of the function f defined on R by f(x) = 8v7 5x