Differential Equation book question

[b2386]

Homework Statement

$$y=e^{-t}Cos(t)+e^{-t}Sin(t)$$

Find the smallest T such that |y(t)| < 0.1 for t > T

Homework Equations

none

The Attempt at a Solution

This question is actually a second part to a differential equations problem, the equation above being my correct solution to a second order linear equation. Now, in my attempt to solve this part, I am stuck because I am not sure what T is (as opposed to t). You have all the info the book has given me so does anyone know what T may be referring to?

 

[AKG]

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, $\infty$)

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, $\infty$)

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

 

[b2386]

Ahh, it makes complete sense now. Thank you for your thorough explanation, AKG.

So basically, now I need to solve for the values of t that make y=0.1 and examine their behavior above and below the calculated values to determine the single correct value of T. From looking at the equation, it appears that it cannot be solved for t analytically and I will have to use Mathematica. Is this correct?

 

[AKG]

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.