Solution of an Ordinary Differential Equation

[controlswhiz]

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.

 

[BTP]

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.