Partial preview of the text
Download Engineering Mathematics Notes: Calculus, Algebra & Problem Solving Guide and more Exams Mathematics in PDF only on Docsity!
— Fivet - order 01 iA Equations - Differential Equation 7 An _qquation cotiaining He cterivaties of ont ov more dependent variables, with respect tone or moe independent — variables eS) oe dy atu. (x2 ty7)de = 2nydy 20 c- dow > de ae By uy = CURSSIFICATION = By TYPE * Ordinary délfeyentiaton Equation ( ope) Fare equations that contain one or more clevivatives of a finctin of a cirgie ( independent) variable. K. on ont Novia be. a -* ‘ Lt id ) at ad Hh *O— OH ar dy stxtly ax x a ot » Partial pittwential Equation ( Pre) FAN equation involving onl} gartial derives of one or more Lwotims at 0 or mot independent variates. x 44 2 2 Ewa + iio : a au ee ,—2U = = ~- Ox .o% x ~UY, dy, : an Sah: i CUPSSIFICATION = BY ORDER tT order ot a differential equation (OPE or PDE) 1s He order of highest derivate in Ye equation CLASS IFICATION By LINGRRITY + An _inth- onder diffeential equation ig said to be lear iP 1? has the form a’y in Standard form rnnneeeeeeeee ee _ Method of coluton Skp ! —— ter y such tt GOED , wok Me differential Pm ! ay =t2L\ gd. 6Q) oes Step t Integrate. Prveboingery Cigar, Jom Jj" Step 9 — Attempt to solve He vembiny equation fy y in toms of x. IE Hs iv possi , we Wave an explicit olution. 12 pe pot pom'bic he soluton ys implily defind by an equation involving x andy. Step 4 Following) Hvis, go back and chk He differential equation fr any values of Y owh that GG)FO. Suck values Ay Wor excluded in wetig Yee iw step (1) and may tead to additonal Solutions — be yond these found in skp (3) 1 & Soe y' # ye” Sol’ first wrk dy sy” ax \€ y#O thi i ial = 170, TNS nas te differential frm | dy 2c “dx ; ui The wartobles hove pen separated « Integrate ! Cr ea J y ae Te atx DATE iN __which k iS constant of ivtgraton . Sovt fr y+ ge yO. to tk @ 4 Solve x*y’ = Ity Sol: The ain 1S sepamble , since we con write ai u ( ; ty r — v= Ba ty x Ig Y#-1 ond x#o. Integrate to obtain | n—t4eyl + x Find 44 explicct solution, Jity] = eke * > ae" whee ase’ Elimvaty que absolut value Symbol py wrihng =| - be ys tae =e * me b=ta Then , | y=-1 the wth b #O ax. Solve 4ne initial value provlem yt eyte® cay zy Sol'n we know 4d He general Dlutin of 4fnis Adfferential equatm is t}= : @ -k Solve fork when yl) #4 { fwan\ 4 bse pits K€. e'-k ; ~ 4 Tho Solution of the initial value problem is I a EE Spel pa? | 4 * KETC) = Ta fated dep AE ___ He cone etont of papirtinally k, aT This 1S yy cepenble — diffranal equation, since, lnttgrate t> obtain In [7 -Gs] > k€ +e RD solve fr Ty take #e copenehtial of both cides of IMS equation te get [T-Gg]= ete 2 Ace whe Az eS Then T-¢ = the = Be oo] ree) Get Be / Now fe onctovtt bk and B mut be detrmindd . Suppore the Kestewaut. aviv at 9°40 pr art immdiatdy measure! Pe body fopuratwe , dlohainiry 94.4°F. BIS onveniont fo lee Fi pm oe tne he Carmiry ont — masuremutl . Then, T Co) -44 7b +B gt | B27 10-4 | Now, Tle)= 06 +26.4 24 qo deturnim k, pe AeA awit, masuremnts. TM liutrant tobe s He broly tonpurotune again at 1:07 pm ark firds it be S1-2°F. cinee 1° pm i $0 minutes after T42pm diy mans teat T (0) = 64-L= OF + Dae4e Tn kK, 2,2 UL 2 AT oe BE 0 ken (Fe) ten fe ae YC) he sumpuratwre finch 6 now Lomplehuy Known as Ziavtig>. Tay = OE + Day gh Cen) €/60 The fime ot darth way the lait tm at. which te beady fompucture wa 0. °F Cit befre it began fo cool). Solve fr te Hme t at which in (ET Ey) T lt)> ape = CPt 204 em et ¢ /e fin gits us 30-4 In(21-2 /2u.u)t /go 20.4 7 © Take the laanthm of fiir “etn + obtain fe, boy fut) Car JF V au] Accordiry h Wa mael, fe tet of death uns £0 In (40.u/ 2.4) In (42 / ta) Which — aporoximntly ~97-& minutes. Death occured —approvinally 52-5 minutes before te firct masurement at 40 pm, whith ver chosen as time pen in He mod. tee puts A murtur ot almt 8°40 pm. Example @- Rodin active Decay ard — Carbon ating Sol ; Radioactive decay is qovumed oy te OPE y'=hy, BY Sepomtion ond inttgnaton (ure & i Ye tine amd Yo ic He initial ratio of Cp 4c* ay. kat in lyl= kt + y= batt Y Not we yoo tHe valf-life HE SPIT fy detwmile k. wh t7H, half of de ongimal substan, is SH pyésent. Thus, Yo ett, O54, gine 0.5 yz IN OF | ~ OHM | - 0-000 11% H 5315 : ; Linear . Eavation 5 oA first-order olifferental is linear ifit has the firm yi + p@y 770 or a(x) 2 4 M iy *9¢x) dx for some function P and g 7 Thee is a general approach tf solving a linear equation. let mAPeeee Oe Oe g&)= PO ee and notice Hrext Gd? pOetPO* . 96x) 96s) ‘ Now _multily iy! tP)y=9O) ww 9G) te obtain E gl) yi + pod aG)y= gle) a) E similarly, we can write tric to ga) y'tA'G)y = Ix) gl) , The let side gf He NM equation is ne Aerivative of QCxy. The Wi evrehiy equahon hag become - A @bdy) « 9&9) ox ‘ WOK we can jitavake yp plot 3040 , we can solve Ws cauation bgv Yi ed ar YO = guy SAG 900 dx 7 G6) t if iy gory © fal at) de FE . c IN r DATE Merred of _ solution linear, ¥' +epQ@dy = 4). ste 13 6 yee “dikereation equation is Fiver compute e pee This ig called an jnregratry factor fr the linear equations Step 1: Multiply tne difeerential equation wey the integrating factor. erivatve of Sep: Write te left side of He resulting equahon as the a the product of y and He inkaratng factor. The integration factor is dlesiqned 4 make this possible. TRE wight sicte jis a function of ust x, Step 4: \wttgvate wotr sides of Wis equation and lve the resulting equation for y, ovtaining the general solution. Ther resulting general solution may S cosCxt) dx) which connot we evyalvated in involve integrals Couch as elementary form. ex-t- Solve’ -y'tY 7% Solin: Tne equation is linear with p&d?l x) a: ‘2 25 Pu) ax gies e* and q&) eX, Tne inttgrating focter 1S Mulkiply 4 differental equation wy e* to oct eXy' te%y > xe™ this is yey = xe \wiegrare his equation +o oltain ye*= Jxe%dx Zxe*% -e* +C Finally, solve for, y boy multiplying his yequation yes poo we te ce? az ex. Save yisoe' =F 3 Goll: ye ee c S Cle) dx In Cx) ze ae fr x 7 0 s nyt? 8x2 or Cxy)! = Ox? 2 .2e oD c VER TE *y -2xt +C => 2 yO224e°5, 67 F * yea” +z] woe! BL tb oy Bets. STHEOREM 4202 1f MC%Y) ond NGKY) are both homogenous and of the some clegree, the function Mdx,y) / NC x,y) ss homogenous of degree zero. r THEOREM 2 If FCKY) is’ homegenus of dese ree in x and y, fay Sa function of y/x alone. ) PROOF: Let ws put y=vx. Ther thertm 2 states pat, if Flay) is homegeneous of degree ze , f(x,y 16 a function of v alonc- Now flay) =f Give) = xfCiv) = FG) in phir the x js now plying He role taken by dX. 7 Homogeneous equation is also sepapsble © Suppose: trot! the cocfbcients M and Non on equation of order one, Miny) dx + NGvy) dy +6, are oth homagenenrs functions and are of the same dearee MA in x andy Py theorems | and 2 ts oige ESM eb oe” ws Vg ber Y2NK , tre equation becomes ni 2M oe wv y+-giv) 20 4 7 4 (4) ax 4 Save Kau equation Gt ~ xy ryt) dx- xy dy © 0 ay oy er. — - Solin; Ler YeN¥> Veaqiy) ov 4 xt KV atv?) ax ~ x4y Cy axt xdv) <0 Simplify Yee equarion,, (ivy tv")ox Hy (vdet x48N) 70 | or Ci-v) de -xvhv #0 separating, HL Nanindts , dv . ; ; ee eee ee oe ei ott imtoo ES x \nregyating «TRE equaton, In] x Cv) oY] = \plet In {xl +v 4 in Wt} zeiw\c) —> x W-lDeKc go fo A xp fir = C Ux ] Ux] i YaDE -c| Fe \ ex. & Solve tre tquaton LY d¥ + (t+ ¥4) dy=0 Solin: Ler xavy, vy? (vay # yav) + (vy? +yt) dy <0 oO vdvay FY) 4 (y2 41) dy 20 separating she voriablec, Pp Nov “Te vy a Ft (aye tl) dyzO , [ we eT integoting the equation, ier ye Eee va In (ay? +1) + 4 Inly! -mco7 « Jy eg fm, fm or y" Cay? +)) =C : ve 4 J* ¥ hus NS z= lwjul 4 fy lyfe 0 5 a [ a6: i zt or ly (ax? +44) ze Vabery | YS ee ee, ee ee ee i ee i ne no ee en a eo ae ae |. Le | 4 sien. aiteerenbine. @) with respect omy) art ascume Of /dy + NL¥Y) : a. Cy logy dx +g iby) * NO*y) dy 9 Thus, Q)= NCxy) -_d ate Wty) yy f M-Lery-Ae—_® +Fivally, inte (5) with respect to y and substittke He resul in CP)- Te implicit colution of the equation is f Guy)=c. Re & Solve He -equchon Dx (py -2) dx +(x? #2y)dy20 Solin: First | from He fact fat , aM 2x? and an Pe tac dy dx Thuetore , its Solution jy FC , where OF MM = oxy - ox oF AJ a? + dy ox oy 1 Inteqertion. cf beth stels of MCny) with respects po x, holdings y constant yea Fz x?¥~ 9x? +TQ), Piffuentiating | Fweretey amt -quating jp to NCxy) wary T'G) 2x? +2y Va)? Ly Thn, Ty)? y? Substituting Wis of F, P) Fex?y - 3x? +y? Finally , bs WY ~9xt 44220 | ——S DATE £x~ 3 ry > > > SOK HE fAllowing (2x9 xy? ay 49) chy ~ Cary 42x) dy 20 So\'n : Fiesh y from Ye fact thot OM | -axy ~25 9N oy o* Thucfre, is solution is Fe c, wre OF 4? sy? Qu 4B OF = -xty -2x f 7 ra (oka dy Imkgration of byt cides of NGwy) with reipeer to y , Welding < constants yielde Fe ~ $x? ~ 2xy +t AG), Differentiating Fuck x and equating jt 7 MCmy) , = xy7-2yt a's) 22x? ~ xy? ~2y +3 Qa) 25343 Threfye, QW = 5 Kt, ak He desired set of colntions of Fig defined implicit by —g xty? - Bry +Hx4 t3ax2=4c, or Kt x77? ~ day Hex 26 === 
