Answer to Question #320445 in Differential Equations for mimi

Question

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.

Answer 1

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