MATH 202 Exam 2, Spring 2009

Math 202 — Spring 2008 Differential Equations, sections 17-20 Quiz 2, April 11 — Duration: 1 hour Reminder of instructions. READ AT LEAST NUMBER 5. 1. Write your name, AUB ID number, and section number ON THE FRONT COVER OF YOUR AUB EXAMINATION BOOKLET. To remind you, the sections are as follows: Section 17 Section 18 Section 19 Section 20 Recitation W 8 Recitation Tu 2 Recitation Tu 3:30 Recitation Tu 5 Prof. Makdisi Mrs. Karam Mrs. Karam Mrs. Karam 2. You may work on the problems in ANY ORDER in your exam booklet, but please make it clear which problem you are solving on any given page. In particular, PLEASE INDICATE IF THE SOLUTION TO A PROBLEM IS CONTINUED ON A LATER PAGE. 3. Do as much of the exam as you can, and budget your time carefully. The problems are arranged APPROXIMATELY in increasing order of time needed. Take a minute at the start of the exam to decide which problems to work on first and to plan strategy. 4. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. Each problem is worth 12 points, and there are 6 problems for a TOTAL 72 points. 5. If you cannot do a certain integral, just leave it as an integral in your solution for partial credit on the rest of the problem. YOU MAY LEAVE THE SOLUTION TO A DIFFERENTIAL EQUATION IN IMPLICIT FORM. 6. No calculators, books, or notes allowed. Turn off and put away any cell phones. GOOD LUCK! (Remember, each problem is worth 12 points, for a total of 72) cos © ys 1. Find the general solution of 4zy’ + 2y = 44 9242 _ ont 2. Find the general solution of y’ = yer xy’ 3. Find the general solution of (32? + y) dz + (2° + ay + x) dy =0. 4. Find the general solution of y” — 6y’ + 25y = cos5a + xe*. 5. Solve the initial value problem y’” — 3y” + 4y = 0. y(0)=1, y'(0)=0, y”(0)=0. 6. We are given that y, = x~? is a solution of the homogeneous linear differential equation yl + Baty! + 4a *y = 0. a) Use the method of reduction of order to find a second solution y2 of the above differential equation. Please assume that zx > 0. b) Show that y and ye are linearly independent. c) Use variation of parameters (NOT reduction of order!) to find ONE particular solution (yp) of the inhomogeneous equation yf tba ty! + 4e-?y = 27 ?, (\ Msg +24 = oe Rerroslly wth ne 3 make subsk Qeyl%= 4 J Se ve Steg rarlk gly Me DE Ly “2 Ge ther vu! *PPpeare 4 “hey + ly. 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POM Be = Gani (ne nr) = Mryim-2) M,~-S Ma zMge Shes oh \\: hers eo + Ce * | AWberer\iche ts ANre ( \atkfal Condibiers : 4> hes 4p Re™ + Cre Yo Remy 2B OC BE) Le 4"° Ae™ +4Re™ + Of He. dee) whedly Ch +R ml ~A +B + Fo t4HR r4C 2A C= A +8 -_ A +8 = | Be +C = { c=) BR ES = | SR + 4C 2-1 Bac F +2 <i“) |ea\ [5] 4 4 2. asa srlka, & svdboshh {7 yve KOU , 2\_! 2 a) v +(- uy Asay ty vy chak oO. Nisha CAA fe Oz yile Seyi Gy \= es vi - =) apie See, fs aw WK Se ean ee Kore o* za eseslediy ; deo we rm _ Ske Wwe) se an —~ KW ibe de 2 tx Se \,q wo = ~lex (we ony nook oye Wy earn ne we Corll eve \n ele -hy eC) Se \ : nS x Ly = x ~ Os <A Ss we take Pus tax} ( we ve oy teed ' Vhs. wet So Any rae WA seule . qerer\ Vis orAlaxr 8° Check Ub. Ye oh en eh, Cyr Cp.oce " ; . _ 7" L et: chal. 1 * mass Inx ° 3 se, so = x, CX lays o —— x Slax te tn Ce er crude, wei) . ee “an ik L ) Ne e) P ki Wea Cy se alto, _< x ~ rdeeall ere > __ = 2x Flap ee s 420° lay = , het Welker (% ek, revue 6 On HARA hone trades Vrerly wg $4 Moves ) More. Vu, Lo, sah bede Bre ype us Mate = xo ARV at © 48° ra) uv) \nx Oy) ee —2 4 Vy © va “ix ,! = © =) ies . 1 Ca (Oe . a _ xu} +(x mer) U2! = x7 VS ares (© bo do) * lori) ve % & Roo ree di, (e——)) Uy + \qx 04! +0 oy = =~ Inx the, o> Lys ‘ ~ > ‘ L . ce Bx \ jal ‘Os = & ty = * \ ute § xn dy = ~% 8 tage dG?) = Ae tay _\* 2): mA (ilo - 3 | S fu, = ~ Wel & x 3 A