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What are some defining characteristics of the solution curves of autonomous ordinary differential equations? I have difficulty figuring out which of the solution curves below correspond to that of an autonomous differential equation system.

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Hint: Since a solution $\gamma:I\to\mathbb{R}$ checks $\gamma'(t)=f(\gamma(t))$ for a fixed $f:\mathbb{R}\to\mathbb{R}$ and for all $t\in I$, what happens when there are two values $t_1$ and $t_2$ such that $\gamma(t_1)=\gamma(t_2)$?

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  • $\begingroup$ If I understand this correctly, it means that it cannot happen that $\gamma$ takes the same value for different $t$-values but also have different gradient. Because an autonomous system should be horizontally translatable. Is that correct? $\endgroup$ – NetUser5y62 Feb 6 at 14:27
  • $\begingroup$ Yes, but we don't even use that an autonomous system is horizontally translatable, just that is is a solution of an equation of the form above. $\endgroup$ – Balloon Feb 6 at 14:59

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