MIT18_03S10_3ex

MIT18_03S10_3ex - 3 Laplace Transform 3A Elementary...

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± ² ± 3. Laplace Transform 3A. Elementary Properties and Formulas 1 3A-1. Show from the deFnition of Laplace transform that L ( t ) = s 2 , s > 0 . 3A-2. Derive the formulas for L ( e at cos bt ) and L ( e at sin bt ) by assuming the formula 1 L ( e t ) = s is also valid when is a complex number; you will also need L ( u + iv ) = L ( u ) + i L ( v ) , for a complex-valued function u ( t ) + iv ( t ). 3A-3. ±ind 1 F ( s ) for each of the following, by using the Laplace transform formulas. L (±or (c) and (e) use a partial fractions decomposition.) 1 3 1 1 + 2 s 1 a) 1 b) c) d) e) s 2 s + 3 s 2 + 4 4 s 3 s 4 9 s 2 2 3A-4. Deduce the formula for L (sin at ) from the deFnition of Laplace transform and the formula for L (cos at ), by using integration by parts. 3A-5. a) ±ind L (cos 2 at ) and L (sin 2 at ) by using a trigonometric identity to change the form of each of these functions. b) Check your answers to part (a) by calculating L (cos 2 at ) + L (sin 2 at ). By inspection, what should the answer be? ³ 1 ± 3A-6. a) Show that = , s > 0 , by using the well-known integral L t s 2 ± x e dx = . 2 0 (Hint: Write down the deFnition of the Laplace transform, and make a change of variable in the integral to make it look like the one just given. Throughout this change of variable, s behaves like a constant.) b) Deduce from the above formula that L ( t ) = ± , s > 0 . 2 s 3 / 2 2 3A-7. Prove that L ( e t ) does not exist for any interval of the form s > a .
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MIT18_03S10_3ex - 3 Laplace Transform 3A Elementary...

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