Chapter 5dydt=ddt(et)=et=y.Thus, given pair of functions is a solution. To sketch the trajectory of this solution, we expressxas a function ofy.x=e3t=(et)3=y3fory=et>0.Sincey=etis an increasing function, the ﬂow arrows are directed away from the origin. SeeFigure B.29 in the answers of the text.3.In this problem,f(x, y)=x−y,g(x, yx2+y2−1. To ±nd the critical point set, we solvethe systemx−y=0,x2+y2−1=0⇒x=y,x2+y2=1.Eliminatingyyields2x2⇒x=±1√2.Substitutingxinto the ±rst equation, we ±nd the corresponding value fory. Thus the criticalpoints of the given system are (1/√2,1/√2) and (−1/√2,−1/√2).5.
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