Understanding the Definition of a "Solution" of an ODE

Hi.

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 $\displaystyle \phi$, defined on an interval $\displaystyle I$ and possessing at least $\displaystyle n$ derivatives that are continuous on $\displaystyle I$, which when substituted into an $\displaystyle n^{th}$ 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 $\displaystyle I$. Correct me if my understanding is wrong. $\displaystyle I$ is the intersection of the domain of $\displaystyle \phi$ 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 $\displaystyle \phi(x)$ is a solution to the first order (or whatever order) DE $\displaystyle F(x,y,...,y^{n-1},y^n)=0$, state the interval of definition."* would be great.

Thank You,

VN19.