I am learning the series solution method of solving differential equations, and I am curious as to what the rationale is for finding out the solution of the equation about a particular point. It seems like this would be of limited use as the behavior of the solution would be undetermined for the "rest of the space".
As a way to further justify or motivate my question, one set of lecture notes I'm using asks us to find the solution to a differential equation about a particular point, while another set of notes asks for the solution without specifying a point (and this one's solution assumes the point to be $x_0=0$).
Sometimes, for example, in physics, we only care about behaviour around a certain point. A not-so-good but common example might be $t<0$ in a plot versus time (which is always neglected in elementary physics). We might want to examine asymptotic behaviour (i.e. getting very far away, very far back in time, very dense, etc.). Other times we want to consider what happens right before and after a certain event, for example a collision of two bodies in the context of their motion - a mathematical singularity. I think there are almost endless reasons that one would want to evaluate the behaviour of a differential equation (or other mathematical objects, for that matter) about a specific point, especially when it is difficult to solve exactly. For applied mathematics, approximation schemes are entirely vital in an era where even computers struggle to handle the computations we require. Expansion about a point is just one example of this.
As far as pure mathematics goes, I can't really speak from experience, but I would imagine that it is important to know how to do this at the very least, because it is incredibly useful for people who need to use mathematics outside of pure mathematics (i.e. almost everyone). Putting such approximations on a rigorous foundation is very very useful so that we can not worry about applying them to, say, building a pacemaker.