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Homework 1 Solution

# Homework 1 Solution - HOMEWOR K j"-s o.¢ 5 4“(a...

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Unformatted text preview: HOMEWOR K- j. _ . ' " -s+o.¢' ___ 5+4“: (a) Em _ (swag? 4-12z .s"+ my: +/W./5 (b) \$41!) = 4""- (4t+ £5) = 0.5-4.3“. 41‘ + (7.8]; Cat 42’? =__0'\$X£ 0-3155 __ 2+o-J’a’ts F29 9+4" + s‘+¢‘ "- £34444; ‘2. (a) £65): 3m (51‘4- 95") = 2/2} 144.5? + 2.12/ cast 5w = 31351315; 1. _2-L2/_5_ _. la-toﬁ- 1/21: 32-1-3“ ‘ ski-9‘ " 6’4-25‘ (b) 738) =Aa3- 0.93 ca 2:" Fiﬁ): v.43 '_ p.035- __ a.o3:'+a./z—a.a3:‘ “ 8‘14 _ s 2 (5 1+0 = 042 sash-94) 2" - z .ZUZJ‘r-‘g-T LL m9= mzw- mwt =a‘ (ma/f + Ch 0‘) =J. _.._S_.'.____ 5 F69 _2 s'+z.sw1 + :*+u1) __ I (\$’+/3w‘)5. - __ —-.—._...______ (5'- + 251%)(3'4- w‘) W. fﬂ)=(2‘—¢) 1(19—1) = 6-48 . F6) 5‘ - ' 3%): t it?) — (f—T) iH—T) f i _ -—/— e_T-f __ —-T.r r _ . 51 - 5-2. = / fez I 7. _____. ﬂ f‘f): ﬁst... _.2£ I --“r-' ' M» a, ———£‘£——— 3) “90 ‘1 4—» 4" 4 . 2 _ r-r_ «a: = z t. jag-{1’ 453 e ) 5' 4—)0 J 7: 747(4) ‘ -31 . -5‘e'§"+4-z’-e"=7‘+se"’ 3‘ a-n 347: ' - :7 _\$. .3; a. zéc—se *‘m-é-e “we”? 83 “+0 .1- '1. «'4 C“ )J- as, A. g _ 1 _ - 3 =%—l«~f 32—8 *‘Pi- "Te ‘-5‘e"" 190 z“. a' 1 -'§:'I'_ 45" --’-.r I —IS =13.” 71(5 3 Te It 3' e ) 5' 4-» d 77(24) .9. A , 1a.. 5(a=&.~...s-F;&)=Ag—i€‘-’—— /0 t9“ 3-): :90 3(34'0 — 7To verify this result, note that I". == (/O-n/J e" ) - My; (/J—xac“)1ff) _= M {906 fl: 5(o+)=/;::_3F¢)=A:.q '3 =0 3-)“ (5+2): um] = W - Kﬁ) .—.- ms) Hence . [email protected]+)=,&.'n 37%): 14,, 3’ _; 39m 39m (3+2): .— 12 . 5+! / sum—or F: = .- 7 = _ .— 0 ‘7 +5+0 - 5' (s+a.s~)*+azr L_ 54-0.! ‘ 4.:- a?!“ = s _ 1+ (Ho-sf H351 0.?“ (sq-4:)" + 9.846" __ ,_l -.51‘ ' l l I- , ' .- J-(t) .. I e " mail/2‘1- A,“ c 4.”,,aﬂ(z‘v W 41-3.; I \$37 - ?+ 5-2 3369-: 5+3t (b) \$s+2 _ 3 P 3 F5.) (s+t)(s+z)‘7' . 5+1 1' (34.92:? 3+2 326‘) = -ae“t'+IPz‘ e'z"+3 3““ 12. ....___/_.__.r.‘-_£_ _L_ / Fm"— :‘(sz-Ha‘) —' .91 3‘ 32+WL) f‘")=3%(f"zf?*‘"”*) ﬂ . o *"' 2x+7x+sx=o, awe-3, xﬂv)=o ZIS‘XK-ﬂ -3‘z(¢9.-i(o)3 + 7 [3X69 — zap] +3X(d =0 -' (251+73-f'3) Xﬁr)=¢’5+2/ . I m. Il’t): a 8"“"t-i-(b4- (duct) 8"“?!- 3: + 29.5" I Pad-27C =' 5(2‘) , I 169—) =0 5X69 - K”) -I.- 2M3) =/ (s+z))((:) -= / xcs) = —L . 5+2 1a) =e‘” ML)- __ __i --~lJ‘*<—/ +3 1, ' __ r: V .. _.z(r2 '7’ .2 ( ,._,, — r ,..__3_,_ 3e 3* 3‘ I”? - 4- J!'—/ +3 > 1, 7 - ,m— + --— ———...__._____ (3-! "l)an'f { 2. ( r-—-,.._/ -+ -—-———-2”. FL]; e r _ . (\$>/) Laplace tranSforming both sides of the differential equation. weoget :Xﬁ) —— 27') +4 X6) = A a; :H—w‘ Ol'_ . Aw (5+49IA’6S‘) -'= m .+ b Sowing for IRS). obtain I I Aw - b .s = .._........_....._._.. X0 (5+A)(:’+W‘) 55*" __Aw /‘_ 5—4 )+lb "7 431.91‘ 5+4 311-103 5+4- v=(,,+_M;_-_ _/_._ 44 w‘ w - It‘d-d" 5+4 7 4‘44"- :‘Hﬂ ‘ __ Ah) .9 7 aH-wf =‘+w-" Inverse Iaplace transform of ﬂu) givus #9 = I “[ij ‘ Aw ' ~41" Ail . __ Ad? ' 4‘1-21")? + a‘J-m‘ “a”?! alt-0‘ ' . =03- ...
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