Question #320445

Consider the autonomous DE xʹ = x(x − 1)(x+ 2). Determine the

critical points of the equation. Discuss a way of obtaining a phase portrait

of the equation.


1
Expert's answer
2022-03-30T11:18:02-0400

x=x(x1)(x+2)Criticalpoints:x=0x(x1)(x+2)=0x{2,0,1}Linearizationaroundthecriticalpoints:x=2:x=6(x+2)x+2=Ce6tthetrajectoryrunsawayfrom2x=0:x=2xx=Ce2tthetrajectoryrunsinto0x=1:x=2(x1)x1=Ce2tthetrajectoryrunsawayfrom1x'=x(x-1)(x+2)\\Critical\,\,points:\\x'=0\Rightarrow x\left( x-1 \right) \left( x+2 \right) =0\Rightarrow \\\Rightarrow x\in \left\{ -2,0,1 \right\} \\Linearization\,\,around\,\,the\,\,critical\,\,points:\\x=-2:\\x'=6\left( x+2 \right) \Rightarrow x+2=Ce^{6t}-the\,\,trajectory\,\,runs\,\,away\,\,from\,\,-2\\x=0:\\x'=-2x\Rightarrow x=Ce^{-2t}-the\,\,trajectory\,\,runs\,\,into\,\,0\\x=1:\\x'=2\left( x-1 \right) \Rightarrow x-1=Ce^{2t}-the\,\,trajectory\,\,runs\,\,away\,\,from\,\,1


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