MATH 202 Exam 2, Spring 2013

Math 202 Quiz II May 10, 2012 Times70 minutes, N. Nahlus & 1. Itani 1 Ae eer (VERY CLEARLY) Section Time..2/.4...._(-2 points if incorrect) fa) (8%) Find Y=Laplace (y) for the given IVP y" #2y'43y= +e)? 5 y(O)=0 & yO) =4 DO not find_y(t). Hint: Expand the right hand-side 2 V_ e 4(0) _ ye) +9sY- Jy(o) +3Y Problem lab 15 15 80 yo St vot Problem 2 5 s 4 (¢ ~ate ae Yv Problem3ab 18 | (+ Problem 4ab =:12 | [Re Subtotal 50 | 4(!)— hy A + os¥ 67. Le et protiem s 10 | tO) <" ($-s)" Su Problem 6 14 | LV2 Problem 7ab 14 1c Problem 8 ab 12 Ve Y i". s¢3) - ao! (2)Ce}) = « 2. @ / ; 2) + +f Totalover 100 , ” is ,) 3 ($-s)" $~10 f ' do! , @) Go!) , ti e4 = se! (S=5)) s-to s% Is+3 4b) (7%) Find X=Laplace (x(t)) and Find Y= Laplace (y(t)) in the system y''+3x'+3x =0 y'ty'+2x=14+5t 3 y(0)=-1, y'(0)=5, (0) =1, x'(0) =0. Just get the 2 linear equations in X & Y; then STOP! __Do not simplify! Bpely Laplace on belk sides, wf bok eyuabhors- #7. Ss 4@) -y {coy + 3 (8X ee) +3 xX =O oY % yo) -yle) + SY - yoy + aX = ee s¥ 46 -9 + 3sX -343X 20 £¥,¢ 3 € MeleaX- lL. % YC) + XGs8) = Bs Y (2xd)e XG) - 1+ £-s+t / 2) (5%) Apply Laplace Transform to change the following IVP to a separable DE: _ then STOP! ty’ +y'+5y=0; ypO)=a & y'(0)=b X (byt +9'+5y) saalaes xe Z| StL / (a 1") Z (. r\¢ s? J. sye)- ye) (1) (7V- Sy @) - yt) # s¥- ae 2 Y=o CHOY) + s¥- a +50 U “SY a+ Y¥(ses) =o Y (s+8) - st dy dx dX- 32.4¥ ¥Cs +5) ~@) 3a) (12%) Find y(t) if its Laplace transform Y is 1 s2(s? +4) s+8 @ Y= 3 © s2 445429 F(s) by using Laplace inverse (——) (complete the square) Ss S ease tes | | | ' +f te fst@ _ Ste | $+2) st \ | | al _ f (brat & fot ee as =f “st Qt 6 \ S+3) i (svayras- gt be > e cos5sl a7 9056 L § 3b) (6 %) Find y(t) ifits Laplace transform Y is roj- (s+5) o (s +5)? rr Gs) coal | | | | | Oe saber” | | (6" derivative) — 8 y (4)= S & cosit 4a) (7% a Find f(t) in the equation t [uo r0- t)dt= i G= 2) dt (Hint: Take Laplace on both sides.) x (sc (fo. fe- 2) de) = f ( z(t)" de) L (fox Y= £ Cex e” Fis). Fin = #1 30! a (te) a §3 53! ) = S$!) C30) Fos ONT yf” CRD) Ss ’ f= (@Geny. ee 14) Ab) (5%) Find f() in the equation Se F(s)=In & >) Hint: Find the derivavtive F'' (s))_ AND In (= p= cessesuesseeea Ps(st 2 24 fet —= ; S*)_ 962) Gr lt 2 aS « ¢= sP4l S| (ar ai — st \ a ae : dcdx pe e © sea" Sten * caf -CosXx| do 2 | (-sin E+) = ie & ° ee : PQ = ak t 5) (10 %) Determine the right form of inthe DE y® — y® =4x+x?e* form o! y, y bd (Do not_find-the constants A, B, .....) ~% ir qe Gut ute (4) Q@) 6 an wy g's: Ps, | =O m,=0 Cdovvle coot ) Mm =° ae ya ot l=o ,mgcl My =| =-l Au.=% F ye ® | a. 3 a Sys By go Axe) & (Cx20xt£) &™ = (0x38 x) + (cx! en x24 Ex) e” ( ed 2e* e*+e™* Note: The integrals are reasonable: multiply up & down by e*) (Careful: We have zero y’) 6) (144%) Solvethe DE y'-y= a : a oe " \ Ro re er+e = Hd ua an (a _ ia |? & a 2= | yr Peat de Je” « se" ee77 e% eo ee) a x = a dee e*_e-~ ox => i - é WwW —————— 7a =) aes ete” erl 7a) “(7 %) Find-the general solution of the Cauchy Euler DE x°y +2y'-y=0 (Hint: m=1) we ny! -4 =O wi aden +! “try yex m(n-*) (ma) +m —-1=0 a" a m Cab dnts) vote ieaael mtn Bmttan aun-lso S 54 4 8m and Bent 4 Bm -l =O er £4 hapa zee = 2 ae C19, 835+ S54 5 = c,™ ot, % lam tC, ula? n. : telax 7b) (7%) Use the substitution x = ef tochangetheDE x’y''—3xy'ty=7e™* +x°(Inx)° toa DE with constant coefficients. (THEN STOP). Do Not prove the Chain Rule_ formulas E de 3 a \ 7 (c 2) bE a a t se am eeey 1g if 8a) (6%) Use the existence/uniqueness Theorem to find all solutions of the BVP yi''=xy''+y'=0 ; y'(8) =8, y''(8) =1, y'(7)=1, Hint: Try kx Le £ “ 24 uf ee a @)=8% J Gal 7% Se Xny uskn u's k uw nee o-nK +ku=0 ona ~ ior kal us Kn 7 yw on u(¢) = e gee uh wr ! Sor YF) u ‘ (@) = 4 @ @) =| jue ¥ cS u"@) =o - — ey = My v7 Sing ls -% are Coakinvovs e©ves g %