Differential Equation book question

The question is perfectly clear. It means find the number T which satisfies the following properties:

i) forall t > T, |y(t)| < 0.1
ii) forall T' < T, there exists t > T' such that |y(t)| > 0.1

So it's like this: first, consider the set S = {t : |y(t)| < 0.1}. Now this is probably going to look like a bunch of intervals, the right-most one being infinitely long. So let's suppose S is like:

(-5, -4) U (0, 1) U (4,6) U (12, [itex]\infty[/itex])

The answer you'd want then would be 12. See because for all t > 12, t is in S. 13 is not the right answer because, although for all t > 13, t is in S, you could choose something lower than 13 with this property, namely anything between 12 and 13. Note also that 10 is not a good answer, because there are t > 10 not in S, e.g. 11. 5 is also not a good answer, because there are t > 5 not in S, again e.g. 11.

Now in your question, S is quite unlikely to be exactly

(-5, -4) U (0, 1) U (4,6) U (12, [itex]\infty[/itex])

but S, that is the set {t : |y(t)| < 0.1}, will be similar. Get the picture?

 

No analytical approach comes to mind, but that isn't to say there necessarily isn't one. The nature of your assignment will determine whether it's okay to use Mathematica or a program like it at this point.