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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.

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    Please consider editing your question with some work you have done and explain where exactly you are stuck2012-11-07
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    @ Jean Thank you for your comment. I will try to think this over as I go2012-11-07
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    **Hint:** Rewrite your equation as $$x' = x^2(1-\lambda x)$$ and see what happens in the _phase line_.2012-11-08
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    @ pragahava I actually had that in my mind, that's how I came with the lines x=0 and x=1\{\lambda}.2012-11-08
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    @Klara What can you say about the cubic $x^2(1-\lambda x)$ as $\lambda$ varies? How will the sign of $x'$, and hence the nature of the critical points will change?2012-11-08
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    @ pragabhava Thank you, I added my thoughts up. I still have ways to go.2012-11-08

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If $\lambda =0$, $x'=x^2>0$ then IVP's solutions blow up for $x_0>0$ and are bounded for $x_0<0$

If $\lambda <0 $, for $x_0 \in [1/\lambda, 0]$ solutions are bounded. For $x_0 >0, x'>0 $ and $x_0 <1/\lambda, x'<0 $ all the solutions blow up.

If $\lambda >0 $, for $x_0 \in [0,1/\lambda]$ solutions are bounded. For $x_0 <0, x'>0 $ and $x_0 >1/\lambda, x'<0 $ all the solutions are bounded.