Exercises 3.2<<<•>>>>>•<<<<<0a/bFigure 3–A: The phase line forp0=(a−bp)p.Putting this value forkinto the equation forp(t)givesp(t) = 1000e(tln 3)/7= 1000·3t/7.To estimate the population in 2010 we plugt= 2010−1980 = 30 into this formula to getp(30) = 1000·330/7≈110,868 splakes.11.In this problem, the dependent variable isp, the independent variable ist, and the functionf(t, p)=(a−bp)p.Sincef(t, p)=f(p), i.e., does not depend ont, the equation is autonomous.To Fnd equilibrium solutions, we solvef(p)=0⇒(a−bp)p=0⇒p1,p2=ab.Thus,p1(t)≡0andp2(t)≡a/bare equilibrium solutions. ±orp1<p<p2,f(p)>0, andf(p)<0whenp>p2.(Also,f(p)<0forp<p1.) Thus the phase line for the given equationis as it is shown in ±igure 3-A. ±rom this picture, we conclude that the equilibriump=p1is
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