Partial preview of the text
Download Differential Equations (engineering math 3) and more Study notes Differential Equations in PDF only on Docsity!
Arab Academy for Science, Technology and Maritime Transport College of Engineering and Technology Department of Basic and Applied Science MATHEMATICS (3) BA223 ay dy + 7h¥ 5 6y = cosh 4x dx dx |) 4s —5)-9 s((s—5) +9) {i L)/xe™* cos sxax}= U(t=a) Prepared By: Dr Karim Shady (ID Trigonometry q) (2) 3) (4) 6) (6) (7) (8) (9) cos” §+ sin? @ = 1+ cot? 6 = cosec”0 sin 20 = 2sin 6 cos@ sin 66 = 2sin 30 cos30 sin 9 = 2sin® és”. 2 2: cos’ 9 = st + cos 20] sin? = sl —cos26] cos” 0 —sin? @ = cos20 — =I = u y=cot u lea 2 1 , =sec"u = u j uvu?—1 ’ -1 ’ =cosec"u y= ST d uvu’-1 y=e" y=eu' y=a" y'=a'u'ina y=Ihu y'= ayy u =log.u = i u’ i y= 10g, y cn ie y=sinhu y =coshu.u' y =coshu y’ =sinh uu’ y = tanhu y’ =sech7u.u’ y =cothu y’ =-cosech’u.u’ y=sechu y’ =—sech u.tanhu.u’ y =cosech u y’ =—cosech u.coth uu’ 1 = inh” — au’ tps anes 1 = +! f= au’ y=cosh"u y dural 1 = tanh” = u’ y u y Ee 1 = th” ‘= uu’ y=coth'u Y= +1 ’ -1 ' y=sech"u y= =U uvl-u y=cosech"u y= sue cig uvl+u’ u 0) fu'au - n+1 +C , Provided that n # —1 n+1 @) [du =Inju|+-c 6 fetau=2"+c ® Jeoosku du = 58 +c © Join ku __cosku | | O) [tank du = BEC U) [ cot ku du _ hfsin ku] |, 8) [secku du = MBecku+ tank, ¢ h —cot ku 0) | cosec kudu _ lnfoosecku-cotku| | ¢, tan ku 0) | sec’ kudu= +C cot ku a1) | cosec’ku du = — +C (12) sinh ku du = a We sinh ku (13) Josh ku du = +C a4) | — =sin™ ac ° fon, > oa ste (16) na dx = In|F(x)|+C (17) Saye = 2,/F(x) +C ZG , veel chapter 1 — Huh. Ditéerential eg 5 ee aie > Difeetial ee ee . ae Ca"). Ste wh 2. ob BE dx Z = Fm) JZ +32 fy 2) Tun DE Py 24 DATE J | Se Janke XA—> * ov / =— J Vey dy TE @ de re aly. £ +c Eagle: Wyo? (Su-4) (4 G)zo \ J \ = 20-4 de eg J dy = fa mav doc ( J € ed = er- Ute fth= da \ x" Yoorc| ”_ Y(5).2 0 “ o- gn|s*ys\E+c| Q2 &nlas-acte \ t C=-4 \ecaar Equabion . < agmuo] Pomas [BEF PG RCACIE ¥ geacedure Ot sowim i {Evaluate the integeahiag fade A Q) souhon = Any. | u~ Qader Real -r—_fF Pred ze n ar ae 2 eee AQMnrne ONe 4 aye = = A 3) | tan x72dx = \ Ysectae- 1 dx = tn K-x4+C., W) [fiscca da = dn lsu thwnl+e. Zz st 4 Lan = 0s'¢ -sdubin’. eT: @ (%) = cos We dale F diaiell Sec 4 M> Ser secaey = | dec Costa dic 2 = = \canda = Sande ~ aa ones: ca | Sc0r = sdamedc le ¢ PAGE .f 4 DATE off e c enuf + Py OO 4" whe AIS ay) real No: >| => depardble eg _ Lf cE t= O =>» dinear q- % Procodume of soul L) divide whole a by Gy") 7 In Fi “alias ieee In at Bas, i cde 2) Suloshituk, these values in He ey « Yy\ xe C=) Viwtas yay? 3 og oe az2s ee a3. afge” SS JS or J =— Zz ava 32) - dy 28. 4 er = ts ie tt Sele ae [iam sig He m2 Se SONG RGl~ 2 atx)=-5, ee *2 0. a* | _Chuylor 4: Bus” ) — [eer ee [hits 204 —> well th |e : SF ene o — = (3). ve emacs Ly soliton + DO NOT WRITE INTHIS AREA. rishi penne PS ajo fot “#3 J) —> one Haat eT : 2 —3 _La- Getal Gem pour equal. ex a => gewal Pe GAG APT ee CE ee ae ee ae i a: ee 2 _than, subsbilae Gly 1) wake ee and solve Fas ______Spanalllt_equdtian - a DO NOT WRITE TT AREA TE) anwar Equiliors. ee fs gerne colin oi Ms cee ae — @- suns ee = Sp Ob) doe re \ DO NOT WRITE IN THIS AREA (Total for Question 5 is 10 marks) .e y {AINE AE OE 0 A 0 Oa P7 4317401528 Une se | TH) Bernoulli's Cuabien: | sax fea Bac “a gawal (ere Be ib hee o = inate. otherwise = Bernoulli 4 - souhions if *2al,, Pedy 2 QW (E> Givtle while ea by Ope =“ ae dh ze ger ema 3- Sulbshtule Hoce yalesin te q: 1 SB 4 Le) 2 = Qe) axe Ci-nJ u- ¥* (IM) “> SE seGq\2 = (Naty) . |e ne G- Finally > Solve asa line HL Hae = SaGO2 ler | Gad) ber VG, 4) dy =o Ode 4 _ 9 2 - 2 Erol Soldlan y U(%Y4)a€ (a4) — ( A (aay) ar + FCQ) > SW (wy) + 9) Men, compre 2 Sup ly) & Ox). | Fieallly WU Cx,y) = total withotl Repti 2 © aes ed col Feciatls ot ae + Cou" + by! + cy La) ew CS ‘Bon Foc Parez en Yond Seand ocdey ODEs . ano _solub'm fc seam! oder ODES or hom 14 engog ‘eg: Solubion » —] cere (20) _ Assumed 4p : ae r ee: a PS ae “ant | Ante el Cx the tC Ax Qe “feted O¢ Cor OFC fas aso) + Bsn Ca) sinh (wx) o¢ cosh Ga) 1 ae tug be siighive d * : eis ay ss oo ax 5 Sth 6 Se az 5 é : aa & a \ : PAGE ~~ aad _ ‘| DATE oe 4 o\vin 6, Define de . . s a |@ qok oy if ie ® : . DE. : a~ so _1O, Cauake coefkicionls w- —— = —<—<—————————— | £ le # 4+ bu “+ VW = feos e — a) = o- 4: yi iby tly eo dh 4! m* pec 2 a” an m+ m+, =o at Cm +2)(m +3) beak?) = ; - Mm a=>, mM, =3 :4, G ee ce | ! 4 ] ea Va ‘ -@ 45 4 iby wily he ) an £C¢) 2 én = Ae” "= ae eh ute 4 ~ a <% s Yhe® 4 bane + Ue = 4e° : Sx AG Ae aes As Se [ig sac - : i | Ke. 4 —oK r See = Ce + Cone + eX 5 3 Ou + by’ s Boe flex) a U t (\) Solve. for the complengntay fe Yn = Cy Oy + Co Ye am 4 ba + ce (2) Coapite. she ‘Wronskinn aE 4, \ W 2 ws at) He whee W 4o (2): 2a a (@) 4, + Bes On Compile. 40) bk BOs : AG) = ~ (82) dx | IW Blo) - (WG) dee Jw m= |= 0 ~» (m-\)(m4l)eo ad Mel, Maat 2 Wha Ce%s ce*| £m)» coSh@) = x (eX4+ e*) Vio fg i ee + SA $65 a VW/ = Ss é Boda oe en 2er 1 WO Sq = Alx) ¥, + BCA] Yo. A@el afO) i. . . \erlere*) I J wl J =4 oy [At C7) de 2 fee Jag, 2 \ whe ‘ (e+e) Je 1 (or BG: { uel boc ie ay be = fel \ Tx > -qle i rxlic v Pose [ae )44] (oath Cty (e7 
