Consider the autonomous ODE $$x' = x(e^{-x}-x^2).$$
- Determine the stability at the trivial equilibrium $x_1 = 0$.
- Show that $f$ has exactly one equilibrium $x_2$ in $(0, 1)$. Determine the stability of $x_2$.
- Use the Newton's method to find an approximation of $x_2$ with $x_0 = 1$.
My question is in the second part. For the first part, we let $v(x) = x(e^{-x}-x^2)$. Direct computation shows that $v'(x)=-3x^2 + (1-x)e^{-x}$. Hence, $v'(0)=1>0$. We conclude that $x_1$ is an unstable equilibrium.
Next, I try to show that there is at least one equilibrium in $(0, 1)$ by showing that $v(0)v(1)<0$ while $v(0)$ and $v(1)$ are not the roots. However, how should I proceed to show the uniqueness of such an equilibrium?
Note: I also have no problem in part 3. I just include it for the sake of completeness of the problem, or in case someone suggests a search for $x_2$ by numerical methods, which is not the purpose of part 2.