MATH 202 Final, Spring 2006

Not To Be Taken Out P-conme Reading Room Math 202 — Spring 2006 Provecse Maksle 5 Differential Equations, sections 1-4 Final exam, June 7 — Duration: 2 hours JUST A MINUTE — PLEASE READ THE INSTRUCTIONS BELOW FIRST 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 1 Section 2 Section 3 Section 4 Recitation Th 12:30 Recitation Th 2 Recitation F 10 Recitation F 9 Ms. Jaber Ms. Jaber Prof. Makdisi Prof. Makdisi 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 PROB- LEMS ARE ARRANGED ACCORDING TO THE CHAPTERS COVERED IN THE BOOK (i.e., chapters 2, 4, 6, 7, 8); just because a problem comes later does NOT mean that it is harder than the earlier problems. 4. Explain your steps precisely and clearly to ensure full credit. Partial solutions will recejve partial credit. Each problem is worth 12 points, and there are 10 problems for a ‘FOTAL 120 points. 5. Uf you ccunot do a certain integra, just leave it as ai: integral & your sciution for pactial credit on the rest of the prcbies. 6. No calculators, books, or notes allowed. Turn off and put away any cell phones or bocuars. GOOD LUCK! (Remember, each problem is worth 12 points for a total of 12) 1. Find the general solution of (a? + A)y! ~ Qey = (8? + 4)(* +1). 2. Using an integrating factor j, find the general solution of (2a +1) cos y dx + ((2" +ax)siny + tan y)ay = 0, [No “canned” formulas allowed for pt] 3. Find the general solution of yl” — Ay" — 2y' + 20y =e. | continued on Feverse whe 4. Find the general solution of hl2 4g? yy" = ; wyty 1+Inz [Be careful with variation of parameters. You may write down a system of equations for ui, and wu, directly, but you may not use a “canned” formula for the solution of these equations. ] 5. Using a series centered at. 2 = 0, find the general solution of (1+ 27)y" — 6ry' + 10y = 0. (Hint: the two fundamental solutions w1 and y are both polynomials.] 6. Find m so that the substitution y = 2™u reduces the following equation to a Besse] DE, and use this to find the general solution of the following equation: on , » tl wey” ~ Bary! + («" + zy =0. 7. Use Laplace transforms to solve the Volterra integral equation t f(t) ~ i (t~ re" Frjdr =e. re 8. Use Laplace transforms to solve l. ifi<3 yl! + 9y = we : . ‘ #—2, ift> 3, yO)=1, y'(0) =0. Suggestion: define g(t) = 1, if t<3 ; and first compute the Laplace transform #-2. ift> 3, G(s) = L{g(t)}.] 9. Find the general solution of 2 4 0 X=] 0 -1 -5/x 0 1 4 [Please leave the answer in COMPLEX form; exceptionally, there are no points for converting the answer to real numbers in this exercise.] 10. Find the general solution of [Reminder: te“’I + eM is a particular solution if (A — \J)E = 0 and (A— DM = L.] otra , P= x, daly Gey 5 Peady Tae _ Go Aho The in ae Que 3 ts pos e . e ved e NN seaq - \n Or*r4> \ fo At = oe MX e ery |, Mati Rermalire aon . ye ee vy = Yor ~ ¢ a ey ) x e\ Serbeh; o>) = Ee ween} = » Yor u . a4 bo \ dy -\ (Gan be coo ~y \ Gur Zev ced ye \2\ ogy by Ma: wane ae sy vhs), P ye Atis a MA Xz oN iw e werk My = Nx ’ yd vee, — (xe) Sy pot Bed Q a Me ( . ° | 5 ; ex) “oy Iske poo POD & MW. . a yy =O ; alge Ba VC AUS xr) So \ _ 2 fory = Co J sf By eer ee htt tm 4202 © Tey Eek 22 25 ede; wer Bd ©)?- 4(-B 20) +20 te. 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