Consider the differential equation in $\mathbb R$: $x' = x^2-\lambda x^3; \,\,\,\,\,\, x(0) = x_0; \,\,\,\,\, t\geq 0$ where $\lambda $ is a parameter. For which initial conditions is the solution bounded for $t\geq 0$? For which initial conditions does the solution blowup in finite time?
I notice that $x=0$ and $x=1/{\lambda}$ are boundaries for solutions. I think as long a $x_0$ is in between the $0$ and $1/{\lambda}$ for ${\lambda} \ne 0$ then the solution is bounded in between there and it cannot escape.