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, | ~ [— +|Ys fGQd) is ac given. function __D|_ Dependent variable eee “Differential paren ewe Combai, Independent variable | Difkerentidble metficic gli iiel a’ Contato} ____*| Order : Higher order differential efficient contain in differential equation js_said to be order of Differential equabons 2 Degree: Rwer of higher order differential equations . Ch Lifnidicl utione , differential cucbficiek aust leis be in. Geypamial) Write order and degees of given differential meficients | ry (Cay \, 3(_wy\ + hy= Q Order = 2 5 Degree = 1 ; i dz J Ley “599 {sey - aay + 3y =O Ovler = 2; Degrees 1 Cs) Coby ys 32k ty Ws 2y= x =0 Order =3 ; Dewees 4 » : TANG RAED, | : 2 he 2
od y = 2A sin 3% — y= 2axr+b dot g des vi « fad) fae Lay = 20 re 2A sin +8 2 xX d?y 0 dy (x 2) dy Ji dé 24 dx? dx? edhe] de / dt fad : — == dy + dtew’ dy 2 24 ) ly'> 2cx+2eJe Liex® du wih det (fit = y"= 2cx+ 2¢-4e —@ = ig + Go?) dy =/24K dydy = 20 pm dx love 7 m =(_dy wa diy \—2 vy + -x)-0 | iia =Cc-—® dx dx? dic? Ea = dy xd 2x dys Gady =o | dx + dit de® dé C4-a7) d’y - ax dy - dy <0 dat du? dx. — ———— Tees ba 7 ii Sela a ce Ct nl _Salution 2s Re hile aes Pd gin ining al x gm tales rere + | Required quai ny = = cam cag ee ep. ho bya ¢ a) dx 2a: a a a £2 2x = 4ay - hace + Again differentiating wrt x’ - x fray — 2% +: Cofebac,)=0 dy ~ oO] de? ___+ [Diferenioking bah cides tart.” yteo ] | 2x- trai dy —2¢) =O . ll dx xb | i. Coed 2a_dy - GQ= dx Ice 4. ‘ Sdution: | J 4-00 + J4-y"= a Gey) fuk x= sin@s y= sin @. A => | V4 srt, + [4- sit =0 (cin Gr-cin& ) | os 81 + Os O& =a Coin om sin€.) tos_ 6:+82 os 8, ape, ie o2 \( as pfe\ u Be alan 9220 aap R ~ 2 Gos 9) 62 = A Sin O1- Oo 2 2 6-82 < 6m 7 = 4 => ton 6 4 : Biz 84 a 2 a tLe a Mot wit F Ss § Page No YOUVA Types of _Differental Euations a ; Vaviale_/ Seperable Homageneous EE kee Differential equations Differential equations .- Differential equations «| Noviable _/ Seperable differential equations General form: Geneval_selution : F000) dau + 9G) dy =0 f £O0 dx _+ f q Gy) dy = Olx Gry acty Guy 20 s — Chey “dat y Go Jdy= o—® 5 4 log. Ctx) GtyD=c, ___[erentioting by _ Cex) Cary 2G) he ty Cie) fl ° _ boy Carol) + leq. Ctty®) = 20« Cad) Chey? em Cry?) i - xX dx +_y dy =0 log Gtx®) CoryD= 7 dex" dag? (C140 Crt yd" _€ (x! de dif dy = Chto) Cir y*) =k | J tot ] ify? ary ; 1 [ x_dv+ ify dy=c A 2 dt 2d dey" a is Al k : | ' J Tce 3. | a Slution : C 24 sinz) - dy = —@S% fide 2=0 yed ; a Cyt) dx log, 2 +109 C2+ sin 0) =C ) b _ I dy 2 (teen \ de log. 2 + ogg 2=C 2 4 } yet 2+ sinx_/ C= 2 log 2= loa, 4 v f dy ~ =f sn de vl i _| i yr 2+ dine log, Gut) by Greine li | log, Cy+4) = ~log Carsinx) 4¢ “Rus E log Ges ly (24 sinx)=c log, Cy+1) + lon, Colin =). log, 4 log, Gt) + lye = = log, 4 : los, (y+) = log, + yt it Per Sz i ome ous] -E— Cae) Sengont _ = F_= | tenet of Nema Hep) + longiht Bw Yu. 4s = N Subnermal 8g Find the equation of curve passing_through G,2) Soe wish: crpatek of kenge — tk pe ee ia! ja Equation of King ent is y= i dy Gx) dx | fuk H=O ita ys fold Mis the midpoint of PQ. y ¢ yoy = =m dy" Y= ay Cx) Ay + =H a 29 du dx 2 ys W-%y_dy ayy da 2 2% A+X-y de 0 BS Ul a _ if dx Mt dy. be dy. Ho, Yu-%y dy \ He X= yi dy 24-1 dy . 0 Yt dx_ = 2x, vi dx A dx dy dy vy, Quy: dy 0) Vide = 2% | de = 2 dy dy J dy { x eY> cm mi nf Azd —__|xXdy 2 dv -v fits 450 = A484 C => BEC-2 | dx d-v" 3 fut v= 1 => 0 =2A48-C => B-C=-2 4 ____|x_dy = 2v= vty Bag; C2-2 | dx. 1-v* \ 4 \ 4 x dk = =, Ayres kg, Grekyet) | A fad \i by oe G3) ( 4 dx] 4-3 ack io. come Solution : 7 = Sinn te ex” MA @=4 _ ! de -d +44) > Ge (anne [ cin3xdx a fe™ dat Sitde 3 3 Jae] a y.@)=0 => O= 44+ > G=-4 fy -aiosmeer a6 é 3 3 J dy= ad 4 fs ax dy + Sette s fat ys —_ sin 3x en + uty Cyn 4 Co q 42 Beate GPRISEICHS Solution + yo u's) = 2kyy 4 ‘ leg. (4. leg, Git) + log 2 | a he ee NBD Tene ote ell : 4 aee4 log, ee : is web [& bi Kd as F ail dy = 3 G41) hee |= lege G4) +C dx / _ dy = 3x dx 43de _ a at X=0 fays a faMdae thy E Ms, Le 3] |_| — fo} | Linear -Diffaential Ejualions E+} General form + oo PO y = QO) ___ FE Thegration factor co: oo | Genal cantons G)G.FD= (Cr) 6) deve I: Greseral farm = s+ (QD 4= acy) y see x= smnxtec solution. te Siege son factoy CIF) = Pucaae 7” . General _sdulion - y » (oF) = = fC LE) G ay) se _ : sy: feat Cane) y_= osx. |e dt = ea) eres fanadz loge REX | da PCa) da Tree =e =_secx| 4° Tnkegration (ar)-e I _ l = Factor Pa ian Geeeal_, Cy) Cx): = Sent» S.SE)-{ Gs de dus Goulet > OG: [an at QQ) dae: dus ¢ Gdubon | Soluien ae re exe ([(mxJotxhic |g dy yew pee T _ co ve | yisecm = } goede 7 TF = =O =e _ Geneval_: G60 = J GOOD dure _ an To omaky het O ice ms) im Aya fs. * Ky f : Yo kyo ey de ye |e Ne ay Ge) ay. (nts. _— OA a ital ly Jot oak i 1 => 0a=ky-0° ee ee ad , es = come A as : Ys Scene A Gren | 1 a oe a ee SOT 4 CURVES (C3 ER ac ' dey Gen) 260 t= Oo) QQ) . Pak. y zt a da aan : eu) yo dy 2 dk a He) i dn = ' od ha | ui a E Ci-n) 1 dy = dt t 7 __! nd co du oe Caen ar a 7k ati n i f\ , | /_—— Morowitr ss ami | 2. + log, 2 Cog2) > du stu = 90) du de ad Ziqz. = Fion 9, - — du , tus 7 y e+ 4 PAC 2). 2 [a Co bts K al dz 441 241-0 _ arrears crea 2 (logz) dx (a2) oH fs 1 tae Po Ae Cog 2) = =U | FS , eee a (gz) Ce nae Geneval_: ‘1 eli! 3 Awe =. 1 dz = du ag Ba xt Z Cogz)" de de Fcalinn Differential equations (dy - dy Cems e Jeino dx a de a TE yl Ss | an 2 Seatac aie Ce’ eo” oT ie -@") dy Be er pel dx Wo 24k | arora. dy. a ia _— 224 Se ts sane os etc] f y= -e 46 ah ey eis.) Chet ee ee = yp) te a es dx x a Ss 02, 1, eye ly Nl ty oe a P| gee _ — cs x” J oe 3 =. -& “x. #- = * pete ne Ceres Clog cx) G Fa 7 ee = log a- 1 es ae il ea (log cx)” a a
