I'm reading through Paul's online math notes on differential equations because I'd like to have a basic grasp on the subject (I'm interested in physics ...)
There's something I don't understand about initial value problems. I get that the solution y(x) to a first-order differential equation with a given initial value constraint might not be valid for every x. But what I don't get is that the author claims that we should only see the interval of validity in which the x of the initial value constraints lays as the interval of validity.
I'm getting these claims from this page, at the Example 1 section.
I just see no reason for this claim and I was wondering if someone could offer a mathematical explanation of why we discard all the intervals of validity except for the one in which the initial value lays.
( I could understand it from a physical point of view ).
Thanks very much in advance, Joshua