393_pdfsam_math 54 differential equation solutions odd

393_pdfsam_math 54 differential equation solutions odd -...

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CHAPTER 7: Laplace Transforms EXERCISES 7.2: Definition of the Laplace Transform, page 359 1. For s > 0, using Definition 1 on page 351 and integration by parts, we compute L { t } ( s ) = 0 e st t dt = lim N →∞ N 0 e st t dt = lim N →∞ N 0 t d e st s = lim N →∞ te st s N 0 + 1 s N 0 e st dt = lim N →∞ te st s N 0 e st s 2 N 0 = lim N →∞ Ne sN s + 0 e sN s 2 + 1 s 2 = 1 s 2 because, for s > 0, e sN 0 and Ne sN = N/e sN 0 as N → ∞ . 3. For s > 6, we have L { t } ( s ) = 0 e st e 6 t dt = 0 e (6 s ) t dt = lim N →∞ N 0 e (6 s ) t dt = lim N →∞ e (6 s ) t 6 s N 0 = lim N →∞ e (6 s ) N 6 s 1 6 s = 0 1 6 s = 1 s
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