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

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

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

View Full Document

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

View Full Document

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

View Full Document
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

## This note was uploaded on 01/10/2012 for the course MATH 2233 taught by Professor Binegar during the Fall '08 term at Oklahoma State.

### 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
Ask a homework question - tutors are online