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# Class Note for MATH 3331 with Professor He at UH 2

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COURSE
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This 19 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
Linearity 1 Linearity af lagXS a fs b gs Math 3331 Section 19470 Lecture 19 March 9 2009 2 19 Reality 2 Reality ft real gt f3 real Consequence ft complex gt Re fs CRefs Im fs Imfs Math 3331 Section 19470 Lecture 19 March 9 2009 3 19 Section 52 I yrs w Derivatives Ys yts 3 Derivatives y lts ms y0 lty ltsgt 82Yltsgt sym y 0 ykgtlts skis sk 1y0 sk Qy m yltk 1gtlto Ill Math 3331 Section 19470 Lecture 19 March 9 2009 4 19 Proof 3 Proof 3 for k 1 Use partial integration fuv dt M fu v dt T T T feistyt dt 6 Styt 8 6 Styt dt 0 0 o 6 5TyT y0 T 3estyt dt 0 For T gt 00 T 6 8TyT gt 0 f0 6 8tyt dt gt Y8 gt yX8 8Y8 210 Ill Math 3331 Section 19470 Lecture 19 March 9 2009 5 19 M ultiplication by e F8 fts 4 Multiplication by Get Ce C ectfts F8 0 Math 3331 Section 19470 Lecture 19 March 9 2009 6 19 Proof 4 Proof 4 00 ectfts O e Stectmdt foo e8ctft dt 0 F3 c Math 3331 Section 19470 Lecture 19 March 9 2009 7 19 Section 52 Multiplication by 1quot F09 ft8 5 Multiplication by tk k 012 tkft8 1kFk8 Math 3331 Section 19470 Lecture 19 March 9 2009 8 1 Proof 5 Proof 5 for k 1 Fs 2 f000 6 Stft dt gt ms few dt tfts 2 Math 3331 Section 19470 Lecture 19 March 9 2009 9 19 Basic Pro erties Examles Transforms of Functions Encountered ODEs ODEs with constant coefficients gt functions tkeCt k O12 Math 3331 Section 19470 Lecture 19 March 9 2009 3910 19 Transforms of Functions Encountered in ODEs cont Property 5 gt k qtkecms oo aedxs Property 4 gt ect8 ect18 18 C 1 S C gt tkects 1k k dk 1 dSkS C Math 3331 Section 19470 Lecture 19 March 9 2009 11 19 Section 52 Basic Pro erties Ex m39les L Transforms of Functions Encountered in ODEs cont 1 gt Special Transforms 0 160 CER gt ects o k O C iw gt memo 8 s2 l w2 COSwts 2 Reg 2 82 111 Math 3331 Section 19470 Lecture 19 March 9 2009 12 19 Section 52 Ba c Pr t e EX mles Transforms of Functions Encountered in ODEs cont 0 kOcoz i gt WWW 3 5 15 at 8 04 gt e cos ts 8a262 6at Sin ES a gkw Math 3331 Sect mn 19470 Lecture 19 March 9 2009 13 19 Section 52 Table of C Transforms Table of L Transforms f t 1 tk ect tkect cos wt sin wt 60 t cos 6t eatsin t Jiwen He University of Houston Math 3331 Section 19470 Lecture 19 ft8 1 S k 8k1 L S C k L 82w2 L 82w2 s cv s a2 2 8 002452 March 9 2009 14 IQ InClass Exercises Exercise 523 3 Using linearity and Table 1 m2 4 5s t2s 4 ts 5 1s 2 1 1 F4isii5lt 2 4 5 33 32 E 2 4S 552 s3 provided 5 gt O H Jiwen He University of Houston Math 3331 Section 19470 Lecture 19 March 9 2009 15 19 Exercise 525 5 Using linearity and Table 1 2 cos I 4 sin 3ts 2 cos ts 4 sin 3ts s 3 2 4 521 29 2ss 2 9 12s2 1 s2 1s2 9 253 1232 18S 12 52 152 9 provided 5 gt O H 9 Math 3331 Section 19470 Lecture 19 March 9 2009 16 19 i In Class Exercises Exercise 5219 19 If y 5y e z with y0 1 then My 5340 2tS yIS 5 yS S 2 saws y0 5 yltsgt S 2 Ifwe let YS ys then SYS 1 5YSS2 s 5Ys1s2 m S i S m 5 Iii Jiwen He University of Houston Math 3331 Section 19470 Lecture 19 March 9 2009 17 19 In Class Exercises Exercise 5222 22 If y y sin4t y0 0 y390 1 then letting Y s y s s2 13005 sy0 y 0 ys s2Ys 1Ys 5216 Solving for Ys s2 1Ys 1 m s2 1Ys i fg 32 20 Ys Fl Jiwen He University of Houston Math 3331 Sention 19470 Lecture 19 March 9 2009 InClass Exercises Exercise 5239 39 If y y 2y e cos 2t with y0 1 and y 0 1 then with Ys ys LU y 2ys s2 ys sy0 y 0 s ys y0 2 ys s2Ys s 1 sYs 1 2Ys s2 s 2Ys s Jiwen He University of Houston Math 3331 Section 19470 Lecture 19 Because the transform of f t cos 2t is F s ss2 4 the transform of equot cos 2t is s1 51 F 1 3 s124 s22s5 Equating 2 s1 2Y S S S S S22S5 SolvingforY 51 Ys S s2s2 32s252235 s322s5s1 s2s2s22s5 s32s26s1 32s2s22s539 Iii 1919 March 9 2009 Lecture 19 s 2 Same Pvnvemes M m Lamace Tvansmm J W H 2 Wm M WM WW M mm m KVLuNnvmmzn

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