AEM_3e_Chapter_02

AEM_3e_Chapter_02 -...

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Unformatted text preview: -4-224-224xy-4-224-224xy-4-224-224xy-4-224-224xy-3-2-1123x-3-2-1123y22First-Order Differential EquationsEXERCISES 2.1Solution Curves Without a Solution3.6.9.12.15. (a)The isoclines have the formy=x+c, which are straightlines with slope1.8-2-112x-2-112y3-25-1-22.1Solution Curves Without a Solution(b)The isoclines have the formx2+y2=c, which are circlescentered at the origin.21.Solvingy23y=y(y3) = 0 we obtain the critical points 0 and 3. From the phase portrait wesee that 0 is asymptotically stable (attractor) and 3 is unstable (repeller).24.Solving 10 + 3yy2= (5y)(2 +y) = 0 we obtain the critical points2 and 5. From the phaseportrait we see that 5 is asymptotically stable (attractor) and2 is unstable (repeller).27.Solvingyln(y+2) = 0 we obtain the critical points1 and 0. From the phase portrait we see that1 is asymptotically stable (attractor) and 0 is unstable (repeller).30.The critical points are approximately at2,2, 0.5, and 1.7. Sincef(y)>0 fory <2.2 and 0.5< y <1.7, thegraph of the solution is increasing on (,2.2) and (0.5,1.7). Sincef(y)<0 for2.2< y <.5 andy >1.7,the graph is decreasing on (2.2,.5) and (1.7,).9-2.20.51.7-2-112x-2-112ymgk$%%mgk2.1Solution Curves Without a Solution39. (a)Writing the differential equation in the formdvdt=kmmgkvwe see that a critical point ismg/k.From the phase portrait we see thatmg/kis an asymptotically stable criticalpoint. Thus, limtv=mg/k.(b)Writing the differential equation in the formdvdt=kmmgkv2=km rmgkvrmgk+vwe see that the only physically meaningful critical point ispmg/k.From the phase portrait we see thatpmg/kis an asymptotically stablecritical point. Thus, limtv=pmg/k.EXERCISES 2.2Separable VariablesIn many of the following problems we will encounter an expression of the formln|g(y)|=f(x)+c. To solve forg(y)we exponentiate both sides of the equation. This yields|g(y)|=ef(x)+c=ecef(x)which impliesg(y) =ecef(x).Lettingc1=ecwe obtaing(y) =c1ef(x).3.Fromdy=e3xdxwe obtainy=13e3x+c.6.From1y2dy=2x dxwe obtain1y=x2+cory=1x2+c1.9.Fromy+ 2 +1ydy=x2lnx dxwe obtainy22+ 2y+ ln|y|=x33ln|x| 19x3+c.12.From 2y dy=sin3xcos33xdxor 2y dy=tan3xsec23x dxwe obtainy2=16sec23x+c.102.2Separable Variables15.From1SdS=k drwe obtainS=cekr.18.From1NdN=(tet+21)dtwe obtain ln|N|=tet+2et+2t+corN=c1etet+2et+2t.21.Fromx dx=1p1y2dywe obtain12x2= sin1y+cory= sinx22+c1.24.From1y21dy=1x21dxor121y11y+ 1dy=121x11x+ 1dxwe obtainln|y1| ln|y+ 1|= ln|x1| ln|x+ 1|+ lncory1y+ 1=c(x1)x+ 1. Usingy(2) = 2 we findc= 1. A solution of the initial-value problem isy1y+ 1=x1x+ 1ory=x....
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