I have a misunderstanding regarding a very common reasoning:
Let's i.e. look at the IVP $\dot x=f(x), x(t_0)=x_0$ with $f(x)=(x-1)(x-2)$. Now, for $x_0\in ]1,2[$ there can be made an argument that the solution always stays in between $]1,2[$.
The usual way to reason this is since otherwise there would be a point, let's say $t_1$, such that $\lambda(t_1)=1$ or $\lambda(t_1)=2$ and that conflicts with the uniqueness of the solution since it would have an intersecting point with one of the constant solutions.
But I don't understand this: The constant solutions don't even solve the IVP if $x_0\in ]1,2[$, so why is it a problem that those solutions intersect?
Is the point here that a solution $\lambda$ for the IVP $x(t_0)=x_0$ would then also solve the IVP with $x(t_1)=1$ which is soled by the constant solution and therefor they must coincide?