E1a - Aim/W EXAM 1 MATH 2233 SECTION 002, SPRING 2011...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Aim/W EXAM 1 MATH 2233 SECTION 002, SPRING 2011 INSTRUCTOR: WEIPING LI Print Name and Student # SHOW WORK FOR CREDIT !!! SHOW WORK FOR CREDIT H! (l) (5pts) State the ORDER, LINEAR OR NONLINEAR of each of the following difier— ential equations. a. t4%;¥+%%+%'y=85int- 5rd MAW; {fl/law b.%¥+y%¥—t+1=0. ZMA male/r WWIC‘Mf/d/V /' (2) (1013133) Determine the equilibrium solutions, and classify each one as asymptotically stable, unstable or semistable. AWWS Tania %1fi3) :'ZH(H “OHIO Efiilz‘IEljviLLm Sotufims 0 , 3:0 / 33?, 3:"l (57) 3‘31,» ha)“: 0 ‘ " 1%) > O WEIPING LI (3) (15pts) (a) Draw a direction field for the equation 253/ + (t + 1)y = ZtB—t, y(1) = a, t > 0. How do the solution appear to behave as t ——) 0? Does the behave depende on the choice of the initial value a ? Let an be the value of a for which the transition from one type of behavior to another occurs. - —t . - BIT; 26f, .15: ll 11,310 (r) 2.€ - t fli—esz i lflel/ICQ, tsl, MU):Q £1144 " :qp’ ._.i [n .‘Wl’: ‘ ‘3‘ Li (H if 0k=5(l) >6 1 5(1) < O (L,ch H”) I) X . <2) if 6mm <e", 5’“) > o (3; :J—m ‘ "’l W; f -—’[,‘ ,bH .‘ w If all e ' +323 ztlémo(a€ v "‘95) ":50 fl gatdfi‘em cowbvwded as +-> O. T89. Hm [sellout/e W4 644 Wm Gamice a) 01:3“). (b) Find the integration factor of the equation 153;, + (t + 1)y = 2te“, y(1) = a, t > O, andvalutéizoébcin-Jttf) a ejfctMr: e j iii—d: ‘ at Z Hence, .th) 30/: fee ( 26% 2: at ($3 :7 (2+ cit : t1+C L ll ‘1‘ “it”? “E”? Elm—w => CW6" 7- (c) Find the critical value do exactly, and describe the behave of the solution corresponding to the initial value (10. - —t a) if 61> ef‘} [w H = li‘m(€céf_+(ae—.)€ ) ante-Mid (a) EH<€ £70 fiirm «we: amoral 3 ’ ikéwgflvtjao C“ C l ’ . '5‘ .4 w E55 i were), Ic‘m “to =h‘m "tat :_ O . €90 {WPO DIFFERENTIAL EQUATIONS 3 (4) (10pts) Given the homogenous solution y = Ce‘t2 for the equation y, + 2ty = O. Solve the non-homogeneous differential equation, by using the variation of constant method, .1 yl+2ty=2te't2. I {:1 {a . «t - _ 3905(1636 flit): Cm€ vzt C006 L / ’ Z “ta ’ "t?" ,t 1 <C(t)€t-Z't Cm€ )+Z't'(C(-E)€ l=2te 2. at)? 7. 2t 51 Cg):th Cog—tag L 7”! (5) (lOpts) Determine (Without solving the problem) an interval in which the solution of the initial value problem is certain to exist. (If — 1)(t - 4)y' +y = 0, y(2) = 1- 1 3/4“ game/i) 5 3 O flee) :O~ v _ __L_fi~ ll” " (team) ) (t) F3 Ce'u‘h‘wwems evergwldere exce/dal— 15:) , 4 3 Cf) i5 Cwllwwmé (anagram flier—fle- 4e HwCa Ware eds-ls 6t 9% M WM far with ClOkaZ/x Li I WEIPING LI (6) (lOpts) Using the subsititution y 2 am, solve the differential equation I {1:24-3:92 y: 2 - my '2. Z Z l / Av X +§X V 1+5V -:_, “a -’ V + r“ " =1 / B XV y X ‘LX 2.x-XV 24/ Z Z 2. [IV I; H—3V I V 1+ V S) [L "0“) "x dx 2V w 2V cfcewbwefl’jfl 2/ 21v :1 fix” “TIE?” X V _ x Sfi‘vsdv V i ‘»< "7/" warn/Z) ‘ '“x “i C" ”” g, Hz 1 “ 3 (7) (10pts) Determine the region in the ty—plane such that there is a unique solution through each given initial point in this region for %=m. .. g»; 2. HI; fungi) : (,tija Capffimmfi CW! l "t j .7/ 7/ __L B ’D—i : Ji-(l"'tz'3t 1' ('zj) P v M DIFFERENTIAL EQUATIONS 5 (8) (lOpts) Solve the exact equation (29: + 1) + (2y — 2)yl = O. Safiwfimahzvmm,m “that ykfl)awh “Wail 17LX : M $2x+| %x,3):JM 4" :J‘QZX'H)AX :2 x + X + CM) 1 23—2 7/ ’7/. "33‘": “(i/9):”: a C13) = Jtzfl “2—) A] 33—25 #6 7/ Hence Sock—iri‘ms 1/33) r- Xz—f X + 5343 : C , 4/ (9) (lOpts) Determine if the equation ydm—i—(Zm—ye—ywy = O is exact. If not, find an integrating factor M such that the multiplied equation by ,u becomes exact. M15 Mp! Nam—36,3 /\/x =2. My A? MK . Net—1F exa0+. (ft5)y=Qt(zx—yéy))x <=> fly) «Wu z/ux (zxvyé‘)+/%-2- 1 V IS a {CWEQ'ZW y ,_ So L/xflx 1'; O. L M77'+/k=:o- +z/L L WEIPING L1 (10) (10pts) Use Euler’s method to find approximate values of y(0.1), y(0.2): y’ = —ty + 0.1y3, y(0) = 1, h = 0.1. %w;o) H ':’i . .3 e l+ Ol‘flb Z jLLhIfiO)=*O'I+o.i-lg=o‘[ evi=+o+k=o+o4=m 3,330+fc-eoydk :1'+ 0-\ - 0.] =— i.o\l L iii-hi) ;”0t"l~Oi +04 -i.0l3 2 0,0020%03 {1/ “thaztwlq zaHM :02 flzcyi—Fffifi'y‘)“ : LDI +0.092050‘ .omi : Lo\©Zo%o[ L, (11) (Bonus 10pts) Let ¢(t) be a solution of the initial value problem 2/ = t2 + 005(7Ty), 11(1) = 1- Find the values of y, (1) and y" film):— [z-t &$(Wfl(u)) ill—t 605K I: ‘—1 =‘— O I M”; (W) I = (ta-r CoSGij/ : 24C _ smug)” 7E“ 1[5/ Lfi Cam“ \YJ<$):2-l “ 53‘4LWHQ'W- 541:) ;: 2 v- O-‘TC‘ O :- 2 rz’ ...
View Full Document

Page1 / 6

E1a - Aim/W EXAM 1 MATH 2233 SECTION 002, SPRING 2011...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online