MAT3700. Tutorial Letter 201/2/2016. Mathematics III (Engineering) Semester 2. Department of Mathematical sciences MAT3700/201/2/2016
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1 MAT3700/0//06 Tuorial Lr 0//06 Mahmaics III (Egirig) MAT3700 Smsr Dparm of Mahmaical scics This uorial lr coais soluios ad aswrs o assigms. BARCODE
2 CONTENTS Pag SOLUTIONS ASSIGNMENT... 3 SOLUTIONS ASSIGNMENT SOLUTIONS ASSIGNMENT I wish ou succss wih h amiaio. L E GREYLING Lcurr Tl: (0) Fa: (0) or mail: lgrli@uisa.ac.za
3 MAT3700/0//06 SOLUTIONS ASSIGNMENT This assigm coribus 0% o our ar mark. d 3. a sc Broulli quaio 3d a sc dz 3d L z h 3d a sc dz h a z sc Thus P a ad Q sc ad P a sc sc P sc ad sc z sc sc sc sc sc Us ruls for pos (9) Us o abl of igrals sc a C. d 0 Homogous quaio d d dv Pu v, h v dv v v dv vv dv v C C subsiu back C (7) 3
4 or d 0 Rwri as liar quaio. d P Thus C or d 0 Us h igraig facor 0 d of form M(, ) N(, ) d 0 M N Chck: ad Thus h quaio is ac. f(, ) f ad f(, ) d f Comparig h aswrs w fid f ad f 0 Thus C ad C C o obai a ac quaio..3 d cos 3 Sparabl quaio d 3 cos sc d 3 sc 3 ac () d or rwri as a liar quaio: sc 3sc
5 MAT3700/0//06 d sc 3sc 3sc Thus P sc ad Q 3sc ad P sc a a a a a a a 3 k 3k 3 k 3 ak a C Us o 6 abl of igrals TOTAL:0 SOLUTIONS ASSIGNMENT This assigm is a wri assigm basd o Sud Guid ad This assigm coribus 70% o our ar mark. QUESTION. D D si 5 6 : CF m m 5 60 m m 3 0 m 3 or m A B CF 3 si D 5D6 D 5D6 D 5D D 6 PI si 0 5D si si 5
6 PI 5D si D D 5D si D 0cos si 0 0 5cos si cos si g A B 0 5 (6) 3. D 6D9 cosh3 CF : m3 0 m 3 wic A B CF PI cosh3 ( D 3) 3 cosh3 D cosh3 D 3 sih3 D 3 3 cosh A B 3 cosh3 g 9 OR PI ( D 3) 6 ( D 3) 0 6 ( D3) ( D3) g A B (6) 8 [] 6
7 MAT3700/0//06 QUESTION Solv for ad i h followig s of simulaous diffrial quaios b usig D-opraor mhods: ( D3) ( D) Aswr: Us Cramr's rul: D 3 D D D 3 D D D D CF : m m0 mm0 m wic A B CF PI D D D D D D g A B Ad D 3 D 3 D D D D3 8 0 CF : m m0 mm0 m wic C D CF PI 8 D D D D D D 8 8 g C D 0 8 [] 7
8 QUESTION 3 3. Drmi Aswr: L H( ) S pag 99 of T book () L H( ) L H( ) L H( ) ( ) ( ) L H L H L H Th giv fucio has ow b rwri so ha w ca us h abl o rad off h aswr. s s s 3 s 3. Drmi L s 6s8 Aswr: L L s 6s8 s 6s998 L s 3 3 sih or = L ss Us parial fracios o fid = L L s s QUESTION Drmi h uiqu soluio of h followig diffrial quaio b usig Laplac rasforms: Aswr: 8 " ', if ( 0) ad '( 0). () [8]
9 MAT3700/0//06 " ' sys s0 ' 0 sys 0 Ys ( ) ( ) s s sysssysys ( ) s s s Y s s s Y( s) s s Ys ( ) Us parial fracios: s s 3 s A B s s s s A s B As A B Thus A 0AB B Ys ( ) s s s s 3 () [] QUESTION 5 For a crai lcrical circui h applicabl diffrial quaio is: wih iiial codiios i( 0) 0 ad i'( 0). di di i , d d 0, 005 Drmi h uiqu soluio for h curr, i i rms of h im,. (7) Aswr: Usig D-opraor mhods: 00D 00D00 i 0 i : m m0 CF m j i Acos Bsi CF 9
10 Us h giv iiial codiios o fid h valus of h cosas A ad B: Giv i 0 0: 0 Acos0B si0 A 0 di i'( 0) Acos Bsi Asi Bcos d Giv i ' 0 : B cos0 B i si Usig Laplac rasforms: di di i d i , hus d d 0, 005 d di i 0 d sis si 0 i' 0 sis 0 Is ( ) 0 sis 0 sis 0 Is ( ) 0 s s I( s) Is ( ) s s s i () si [7] TOTAL = 50 3 SOLUTIONS ASSIGNMENT 3 This assigm coribus 0% o our ar mark. QUESTION 7. If 6 5 B, fid h igvalus of B. () Aswr: For igvalus pu
11 MAT3700/0//06 Thus or = 3 7. If A 0, fid a igvcor corrspodig o h igvalu. 0 0 () Aswr: hus W ca ow wri dow h rlaioships bw From quaio : 3 ad To wri dow a igvcor w mus mak a choic for a valu. Choos = w g from quaio (3), ad from quaio () 0 3 Thus a igvcor for λ=0 is 0 No ha w do hav o work ou all h igvalus as i is o askd. [8] QUESTION 8 A fucio f() is dfid ovr o priod b f 0 0
12 Aswr: 8. Skch h giv fucio: (3) 8. Odd fucio bcaus f ( ) f () 8.3 Bcaus i is a v fucio a0 a 0 Th priod =, hrfor L =. No: As h wo igrals from - o 0 ad from 0 o ar qual w ca sav im b muliplig o of hm b. b L f( )si b L 0 L si 0 is a umbr ad ca b ak ou of h igral si 0 cos = 0 cos cos0 0 odd v L Thus f ( ) b si si [] 3 si si... 3
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