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# ELEM DIFF EQUATIONS Math 3D

UCI
GPA 3.73

Staff

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Staff
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KARMA
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## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Adam Crona on Saturday September 12, 2015. The Class Notes belongs to Math 3D at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/201875/math-3d-university-of-california-irvine in Mathematics (M) at University of California - Irvine.

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Date Created: 09/12/15
Math 3D Laplace Transform Problem with Heaviside and Delta Transforms Paul Macklin May 12 2005 Since we didn t get a chance to work on these in discussion here7s an example of a Laplace transform problem with Heaviside and delta functions Exercises Find the solution to the following differential equations l y y H16 W i 5 MO 1 M0 07 Solution To solve this problem we take the Laplace transform of both sides 67w L 9 9 L HAW L Wt 5 w 523 590 NO Y S 6 55 821Y 3 67 6755 3 e77r5 e755 7 Y7L771 32l 332l 32l Now lets recall some Laplace transform rules L Hctft 6 E CSFSA This says that if we see thing with an 6 multiplied it to nd the inverse transform just ignore the 6 evaluate the result at t t7 c and multiply the result by Had We can see this by taking the inverse Laplace transform of both sides of the rule Hctft 7 5 e csFs 1 Hctft7 71 TCSFSgt Hctft 7 c 71 TCSFSgt Hct 71Fs tic 71e Fs Now let s look at our result again 3 e77rs e755 YltSgt 32l T 332l T 32l Therefore we can solve for yt by taking the inverse Laplace transform of both sides m 7 N L W 7 N W 7 N i 7 r1 7 r1 32l 332l 32l 1 1 t 1 457 1 557 COS lt6 332 1 lt6 32 1 The rule applies to the last two terms Furthermore since the inverse Laplace transform of 572 is sin t we can write 1 yt cos it 71 lt67 H5t s1nt 7 5 and if we can determine the inverse transform of 55211 we can nish the problem We proceed by partial fractions 1 7 A B3 C 7 2 mig i m gt 17A8 1BSCS 7 1 32AB3C1A

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