Question

1. Consider the family of differential equations done = y2 + ky + kº. (a) Are there any equilibrium solutions when k =0? If so, what are they? (b) Draw the bifurcation diagram. That is, sketch a graph of the critical values as a function of the parameter k. Clearly label the axes. (You may use Mathematica for this problem, but your final answer must be drawn by hand.) (c) Draw the phase diagram for when k = -1. For each critical point indicate whether they are stable, unstable, or semistable. (You may use Mathematica for this problem, but your final answer must be drawn by hand.) (a) Let k = -1 and y(0) = 0. Does lime-to y(x) exist? If so, what is it? Explain how you reached your answer. (e) For which values of k do there exist stable equilibrium solutions?

Answer 1

1 Answer)- Given that Given differential equation dy y + kytk dx 3 9 where k=0, 0-) Y CY7k) +k? y (y + x) + k3 y²7k=0 at kao / 1+ kuo Y = trk no there is no equilibrium point here. b) we have to draw bifurcation diagram Y S 2 2 y'=fly > k. lengend: Unstable 5-stable N-node c) we have to draw phase diagram for whenk NI dy 3 =ytky tk 3 dx y? - 1 4 - 2 у (y + 3) YED, 8 Y

않을 Y = + 5/2 y - 25 -los Let y=o=1 Es stable. NE >Node su A unstable. y = 1/2 unstable d) Y COC) SC dy ax let k=-5, y {o) = 0 does lim y² 1 4 - 1 Yo- Śy ſ dx = dy Taking Integral Integral on both sides, we Yx -' yx - get 1 =Y у - Ух ZYX 2 yoo Ź you go hence lim x does not exist. from previous solution at k k = 1 NI exist the stable equilibrium solution