Why is it differential equations exist on an interval instead of a domain? I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of the solution to a differential equation over an interval instead of an domain? If the solution to an ODE is just a function, and we normally speak of functions as having a domain, then why wouldn't we use domain? 
 A: To expand on Mark McClure's comment: an initial condition will restrict the solution in an interval.  If you have, say, a domain consisting of two
disjoint open intervals $A$ and $B$, you might take one initial condition $y(x_1) = y_1$ in 
$A$ and another one $y(x_2) = y_2$ in $B$, and have a solution defined in $A \cup B$, but there is no necessary connection between them: the initial condition
$y(x_1) = y_1$ only affects the solution in $A$ and the initial condition
$y(x_2) = y_2$ only affects the solution in $B$. There is no real reason to 
consider them as the "same" solution.  
Mind you, there is a different issue: if you have a closed-form formula describing a solution in $A$, it may also describe a solution in $B$.  But 
I would not consider that a  "real" reason: it's just an artifact of the way we represent a solution.  
A: Well, since the two aren't mutually exclusive you can actually call it whatever you like. What I mean is that an interval is  a domain, so if you say the solution to an IVP is in a domain, you won't be excluding the case it happens to be an interval. Conversely, if you call the solution space an interval it wont be false since we know that, for any IVP, the solution space  is  an interval (if someone can verify please?)
In a mathy way of saying, what happens is that $$\text{Solution space is interval $\iff$ Solution space is domain}$$
There is a caveat, and this is that I can only say this for the ODEs I know of.
