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Could anyone comment on the following exercise from ODE?

For any $\epsilon > 0$, find $\delta > 0$ such that $\ddot{X}+X-2X^{3}=0$ and $\sqrt{X^{2}(0)+\dot{X}^{2}(0)}< \delta \Rightarrow \sqrt{X^{2}(t)+\dot{X}^{2}(t)}< \epsilon$, for all $t \geq 0$.

Thank you!

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Hint: the differential equation says $\frac{d}{dt} \left(\dot{X}^2 + X^2 - X^4 \right) = 0$.

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    Yes, but from this i still cannot derive the desired result, any further comment please?2012-03-27
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    If $A = \dot{X}^2 + X^2 - X^4$ and $B = \dot{X}^2 + X^2$, find an inequality of the form $B \ge A \ge \alpha B$ for $B < \beta$. Now $A(t) \le B(0)$ so $B(t) \le B(0)/\alpha$ if $B(0) < \beta$.2012-03-27
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    This approach works. Thank you!2012-03-28