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Class Note for MATH 322 with Professor Glickenstein at UA

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
Chapter 6 Laplace Transforms Chapter 6 Laplace Transforms Definitions Properties of the Laplace ns39form Applications to ODES and systems of QDEs 1 Definio o The Laplace transform f of a piecewise continuous function 7 defined on 0 00 is given by fs Fs fooo exp s t ft dt 0 Clearly the above integral only converges if 7 does not grow too fast at infinity More precisely if there exist constants M gt O and k E R such that ft g Mexpk t for 1 large enough then the Laplace transform of 7 exists for all s gt k 0 If f has a Laplace transform F we also say that f is the inverse Laplace transform of F and write 2 1F Chapter 6 Laplace Transforms General properties s shifting Laplace transform of derivatives 84 antiderivatives Heaviside and delta functions t shifting Differentiation and integration of Laplace transforms Definitions Properties of the Laplace transform Applications to ODEs and systems of QDEs 2 Propeies of the Laplace transfor o The Laplace transform is a linear transformation ie if f1 and f2 have Laplace transforms and if 051 and 052 are constants then 051fl l 04272 a1 l 0525062 0 As for Fourier transforms the statement f 5 1 5 should be understood in a point wise fashion only at points where f is continuous 0 Since there is no explicit formula for the inverse Laplace transform formal inversion is accomplished by using tables shifting t and 5 taking derivatives of known Laplace transforms or integrating them Chapter 6 Laplace Transforms Defir aitior as Properties of the Laplace transform Applications to and systems of CFDEs 11mm sigma F d leifi lvj Imt dlis wajtc weg seshi ting 0 Note All of the formulas written in what follows implicitly assume that the various functions used have well defined Laplace transforms One should therefore check that the corresponding Laplace transforms exist before using these formulas o s shifting formulas L eatft s Fs a eatft 1Fs a o Laplace transform of derivatives L f s sums No L f s 52 fs s f0 f 0 General properties s shifting Laplace transform of derivatives amp antiderivatives Heaviside and delta functions t shifting Differentiation and integration of Laplace transforms Definitions Properties of the Laplace transform Applications to ODEs and systems of QDEs Laplace transform of derivatives and antiderivatives o More generally L M s s fs s 1fO s 2f O rltquot 1gt0 o Laplace transform of antiderivatives L for rm d7 5 fs ft rm d7 5 1 50 t 0 0 Examples 0 Find the Laplace transforms of sinwt and coswt c Find the inverse Laplace transforms of 1ss2 1 and 1s2s2 Chapter 6 Laplace Transforms General properties s shifting Laplace transform of derivatives 84 antiderivatives Heaviside and delta functions t shifting Differentiation and integration of Laplace transforms Definitior as Properties of the Laplace transform Applications to ODEs and systems of QDEs Heavisideand delta functions t shifg o The Heaviside function or step function Ht is defined as 0 if t lt 0 Hlt1ift20 e as 0 We can calculate that for a gt O Ht a s S o More generally we have the following time shifting formulas for a gt O L ft a Ht a s eaS fs ft a Ht a 5 1 eaS fs t o The above formulas are useful to calculate the Laplace transforms of signals that are defined in a piecewise fashion Chapter 6 Laplace Transforms General properties s shi39fting Laplace transform of derivatives 84 antiderivatives Heaviside and delta functions t shifting Differentiation and integration of Laplace transforms Definitior as Properties of the Laplace transform Applications to ODEs and systems of QDEs Delta funtions o The Dirac delta function or distribution is defined as the limit of the following sequence of narrow top hat functions 1 6U ell th felt felt l 0 otherwise 0 Since 721 dt 1 we also write that 61 dt 1 o More generally for a well behaved function g we have gm 6t a dt go o For a gt 0 this allows us to define the Laplace transform of 6t a as E 61 a s eas Chapter 6 Laplace Transforms General properties s shi39fting Laplace transform of derivatives 84 antiderivatives Heaviside and delta functions t shifting Differentiation and integration of Laplace transforms Definitions Properties of the Laplace transform Applications to ODEs and systems of DDEs Differentition and integration of Laplace transforms In what follows we write fs as Fs o Differentiation of Laplace transforms tft s F s 5 1 F s t tft 3 Integration of Laplace transforms 413 s 2 f PM 5 1 Fm cry 1 0 Example Find the inverse Laplace transform of ss2 l 12 Chapter 6 Laplace Transforms Deiir aitions Properties of the Laplace transform Applications to ODEs and systems of ODEs Applications to ODEs and systems 00E 0 Solve y l y t7r with y7r O and y 7r 1 17r 1 Z 2 6 If 1 6 S t S 1 l 6 0 Let at l 0 otherwise Y 4y 5y 2 ft with initial conditions y0 0 and y 0 0 O Solve y l 4y 5y 2 61 1 with initial conditions M0 01y0 0 where e lt 1 Solve X o Solve the initial value problem CZ t AX err16 xltogtii Chapter 6 Laplace Transforms

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