Understanding the Definition of a "Solution" of an ODE
I'm having a little trouble getting a clear understanding of what is meant by the "interval of definition". I have Zill's A First Course in Differential Equations with Modeling Applications, and here is the definition given of a solution:
Definition 1.1.2: Solution of an ODE
Any function , defined on an interval and possessing at least derivatives that are continuous on , which when substituted into an order ODE reduces the equation to an identity, is said to be a solution of the equation on the interval.
Now, I just don't understand what interval is meant by . Correct me if my understanding is wrong. is the intersection of the domain of and the domain of the DE. If this isn't quite right, please explain it to me.
P.S. Some examples of the form "Given that is a solution to the first order (or whatever order) DE , state the interval of definition." would be great.