Take Clear f g x D f x D Sin x D g x D xCos x D 1 \u00ea 3 Both functions start

# Take clear f g x d f x d sin x d g x d xcos x d 1 ê

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Take:Clear@f, g, xD;f@x_D=Sin@xD;g@x_D=xCos@xD1ê3;Both functions start their race at x=0together:8f@0D, g@0D<80, 0<Here's a plot of the instantaneous growth rates of both functions for 0§x§1.5:growthplot=Plot@8f'@xD, g'@xD<,8x, 0, 1.5<,PlotStyle->88Red, Thickness@0.01D<, Thickness@0.01D<,AxesLabel->8"x",""<, PlotLabel->"f'@xDand g'@xD"D
0.20.40.60.81.01.21.4x-0.50.51.0f'@xDand g'@xDUse this plot and the Race Track Principle to predict a number bso that f@xDand g@xDare nearly the same for 0§x§b.Check yourself with a plot.·Answer:The two functions start their race with f@0D=g@0Dand the plot shows that the growth rates of f@xDand g@xDare nearly the same for 0§x§0.8. The Race Track Principle tells you that f@xDand g@xDare nearly the same for 0§x§b=0.8.Confirm with a plot:b=0.8;Plot@8f@xD, g@xD<,8x, 0, b<,PlotStyle->88Red, Thickness@0.0175D<, Thickness@0.01D<,AxesLabel->8"x",""<, PlotLabel->"f@xDand g@xD"D0.20.40.60.8x0.10.20.30.40.50.60.7f@xDand g@xDf@xDand g@xDare running together on this plot, just as the Race Track Principle predicted. Take a look on a longer interval:b=1.5;functionplot=Plot@8f@xD, g@xD<,8x, 0, b<,PlotStyle->88Red, Thickness@0.0175D<, Thickness@0.01D<,AxesLabel->8"x",""<, PlotLabel->"f@xDand g@xD"D0.20.40.60.81.01.21.4x0.20.40.60.81.0f@xDand g@xDCompare:Show@GraphicsArray@8growthplot, functionplot<DD0.20.40.60.81.01.21.4x-0.50.51.0f'@xDand g'@xD0.20.40.60.81.01.21.4x0.20.40.60.81.0f@xDand g@xDf@xDand g@xDbegin to pull apart shortly after their instantaneous growth rates f'@xDand g'@xDbegin to pull apart.This is in line with the Race Track Principle.B.2) The Race Track Principle and differential equations·B.2.a)If y@0D=starteris small relative to b, how does the Race Track Principle tell you that the solution ofy'@xD=ry@xD,and the solution of its companion logistic differential equation y'@xD=ry@xDI1-y@xDbM,will run close together initially as xadvances from 0?·Answer:Look at the equations again:y'@xD=ry@xDy'@xD=ry@xDI1-y@xDbMboth with y@0D=starter.When you go with y@0D=starterthat is small relative to b, then initially as xadvances from 0, y@xDbis small,and so I1-y@xDbMis close to 1.The upshot:When you go with y@0D=starterthat is small relative to b, then y'@xD=ry@xDand y'@xD=ry@xDI1-y@xDbMare nearly the same initially as xadvances from 0. A version of the Race Track Principle tells you that the plots of the solutions to both equations share a lot of ink (provided you give them the same value at x=0) initially as xadvances from 0.Enough philosophy.Check this out on an actual case:r=0.091;b=500;starter=20;Clear@x, yD;logistic=NDSolveB:y'@xD==ry@xD1-y@xDb, y@0D==starter>, y@xD,8x, 0, 4<F;y@x_D=y@xD ê.logistic@@1DD;logisticplot=Plot@y@xD,8x, 0, 4<, PlotStyle->88Blue, Thickness@0.01D<<D;Clear@yD;exponential=DSolve@8y'@xD==ry@xD, y@0D==starter<, y@xD, xD;y@x_D=y@xD ê.exponential@@1DD;exponplot=Plot@y@xD,8x, 0, 4<, PlotStyle->88Red, Thickness@0.01D<<D;Show@exponplot, logisticplot, AxesLabel->8"x",""<D
1234x22242628Just as the Race Track Principle predicted.