# Solution curves of autonomous ODE

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.

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 $$\gamma(t_1)=\gamma(t_2)$$?
• 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? – NetUser5y62 Feb 6 at 14:27