# Solution of an Ordinary Differential Equation

Hi,

The definition (see attachment) says that f(x) is a solution to
the differential equation if it satisfies the equation for every x
in the interval.

Assuming that I have a differential equation that I want to
solve and the D.E. has an interval $I_1$, and I've
come up a solution with an interval $I_2$,
where $I_2$ is a subset of $I_1$, is it
still a solution to the differential equation? If it isn't, does the
solution still make sense?

I'm new to differential equations and haven't solved anything
DE yet.

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Assuming that I have a differential equation that I want to
solve and the D.E. has an interval $I_1$, and I've
come up a solution with an interval $I_2$,
where $I_2$ is a subset of $I_1$, is it
still a solution to the differential equation?

The blurb could be a little bit clearer. When talking about a solution to a differential equation in a given problem the domain of interest is essential. The blurb implies this but it could be explained a bit more explicitly. So to answer your question, it is not a solution to a specific problem posed on $I_1$. That you found a function that works on $I_2$ would satisfies a different problem, one posed on $I_2$.

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