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a) $\dot x = x(1-x)$, $x_0=1/2$

b) $\dot x = x(x-1)$, $x_0=1/2$

c) $\dot x = x(x-1)$, $x_0=2$

I think that the notation for these problems is causing more problems than the actual math. I plan to solve these as normal differential equations, but I believe that these problems require more analysis than that (Something along the lines of Picard-Lipschitz). Regardless, I'm struggling to be confident in any of my work, so any guidance would be appreciated.

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    The most straightforward way to answer your question is to actually carefully solve all three problems and analyze the obtained solutions. – 2017-02-27
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    Hello - you can solve all three equations with separation of variables. You can also answer parts (a) and (b) by noticing that $0 < \frac{1}{2} < 1$ and that $x = 0$ and $x = 1$ are both solutions of the ODE. Then recall that for scalar ODE's with well-behaved right hand sides solutions cannot cross. – 2017-02-27

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