I am looking for an example for a real first order ODE which has $0$, $\infty$ or exactly $1$ solution depending on the initial $x(t_{0})$ value.
The solution doesn't necessary has to have the domain of the full real line.
I was thinking about something which has similar solutions like the functions on the following picture on $[0,\infty)$.
Is there any good method to figure out differential equations knowing their solutions? If my task is to write a differential equation which has a solution $x(t)$, when does this task has a unique differential equation?
So far I used Itô formula for deterministic processes in these kind of problems. Perhaps it is a kind of "beyond the goal" solution. Is there any better idea?