397_pdfsam_math 54 differential equation solutions odd

397_pdfsam_math 54 differential equation solutions odd -...

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Exercises 7.2 for all α > 0. Thus, fixed α > 0, for some T = T ( α ) > 0, we have | t 3 | < e αt for all t > T , and so t 3 sin t t 3 < e αt , t > T. Therefore, t 3 sin t is of exponential order α , for any α > 0. (b) Clearly, for any t , | f ( t ) | = 100 e 49 t , and so Definition 3 is satisfied with M = 100, α = 49, and any T . Hence, f ( t ) is of exponential order 49. (c) Since lim t →∞ f ( t ) e αt = lim t →∞ e t 3 αt = lim t →∞ e ( t 2 α ) t = , we see that f ( t ) grows faster than e αt for any α . Thus f ( t ) is not of exponential order. (d) Similarly to (a), for any α > 0, we get lim t →∞ | t ln t | e αt = lim t →∞ t ln t e αt = lim t →∞ ln t + 1 αe αt = lim t →∞ 1 /t α 2 e αt = 0 , and so f ( t ) is of exponential order α for any positive α . (e) Since, f ( t ) = cosh ( t 2 ) = e t 2 + e t 2 2 > 1 2 e t 2 and e t 2 grows faster than e αt for any fixed α (see page 357 in the text), we conclude that
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