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: DeFnition of the Laplace Transform, page 359 1. For s> 0, using Defnition 1 on page 351 and integration by parts, we compute L{ t } ( s )= Z 0 e st tdt = lim N →∞ N Z 0 e st = lim N →∞ N Z 0 td ± e st s ² = lim N →∞ te st s ³ ³ ³ N 0 + 1 s N Z 0 e st dt = lim N →∞ ´ te st s ³ ³ ³ N 0 e st s 2 ³ ³ ³ N 0 µ = lim N →∞ ´ Ne sN s +0 e sN s 2 + 1 s 2 µ = 1 s 2 because, ±or 0, e sN 0and sN = N/e sN 0as N →∞ . 3. 6, we have t } ( s Z 0 e st e 6 t dt = Z 0 e (6 s ) t dt = lim N →∞ N Z 0 e (6 s ) t dt = lim N →∞ ´ e (6 s ) t 6 s ³ ³ ³ N 0 µ = lim N →∞ ´ e (6 s ) N 6 s 1 6 s µ =0 1 6 s = 1 s 6 . 5. 0, cos 2 t }
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.

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