Project : Direction Fields and Equilibrium Solutions 1. An equilibrium solution y = c is called asymptotically stable (or attracting) if all nearby solutions converge to c. More precisely7 there is an interval (a, b) containing (3, such that for every co in ((1,5) the solution y = f(93) to the initial-value problem y(0) : on has limmnoo at) : c 7 Remark: Think about this like a ball at the bottom of a valley. If we push it a little, it will still return to the bottom after rocking back and forth. There are ve ways the y' = F (7;) curve might look (at an equilibrium) near the yaxis: A n C D E In which of these ve cases will the equilibrium solutions be asymptotically stable? Hint: Draw nearby direction elds and plot representative solution curves. Project : Direction Fields and Equilibrium Solutions 3 2. An equilibrium solution y(x) = c is repelling if we replace lime-> f(x) = c in the above definition, with limx--co f(x) = c. Remark: Think about this like a ball at the top of a hill. If we push it a little it will roll off. Again analyse the five cases from question 1, this time determining which are repelling