As a continuation of this question, the given answer states that at a point where the ODE is undefined, we apply "analytic continuation" (if this is the correct terminology) to the solution so that the domain of the solution is extended to a point where the ODE is undefined.
However, as you can see the comments to this question, it is still not clear for me why are we doing this ? and when do we have to do this ? I mean as @MarianoSuárez-Álvarez pointed out the second question, that is linked, talking about solutions of an ODE at a point where the ODE is undefined does not make sense.However, as my professor stated earlier, we say that the ODE might have infinitely many solution at such a point, as it is the case in the first question that is linked.
Therefore my question is that when do extend the domain of a solution of an ODE, when do we have to, and when do not do that ?