MATH 202 Final, Spring 2002

fat. Makdes: i N OME VAL Math 202 — Spring 2002 Roe To Be + ” Differential Equations, sections 11-15 Prva p. “Fken Out Final Exam, June 18, 2002 — Duration: 2 hours Jie NAME: Section: 11: Th2, N323 AUB ID#: 13; Th12, B105 14: Thi, B105 12: F2, N412 15: F2, B104 GRADES: 1 2 3 4 5 6 7 8 9 10 TOTAL/100 INSTRUCTIONS: 1, Write your NAME and AUB ID number above, and circle your SECTION. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. . You may use the back of each page for scratchwork OR for solutions. . There are three extra blank sheets at the end, for extra scratchwork or solutions. - Do as much of the exam as you can, and budget your time carefully THERE ARE 10 PROBLEMS ON THIS EXAM. Each problem is worth 10 points. 6. No calculators, books, or notes allowed. Turn off any cell phones or beepers. GOOD LUCK! of OO Use the Laplace transform to solve the integrodifferential equation t uv +2y-+10 [ y(r) dr = 2, y(0) = 1. r=0 Use Laplace transformis to find the solution of the initial-value problem ” , . 0, fort <1 t y” — 4y' + 3y =4 f(t), where f(t) = { t-1 ie > L and y(0)=2, y'(0} =6. e ’ Find the general (implicit) solution of the differential equation —5 y > ve 27 ying. xz 2 Find the general solution of the differential equation x?y” — zy’ +y = = Find the general solution of y’” + 3y" + 9y’ — 13y = 8cosz + 6e”. Find the (implicit) solution of the initial-value problem (1+ 3zsiny) dx — x? cosydy=0, y(1) = | Malla: 2% Spry Ler (Tone I, 20°2) Real exam . rf. Maks: qe U2. . 7. a) Using a series centered at x = 0, find the general solution of the equation: rd (L+27)y” + 2ry’ ~2y =0. b) Bonus (2 points): Express your answer in terms of familiar functions. 8. Note: The different parts are NOT related. 1 a) Given that L{Jo(t)} = Vata find the function f(t) which is given by 8 f(t) = Jo(r) Jo(t — 7) dr. 7=0 b) Given that [(1/2) = /m, what is the value of T'(7/2)? c) Write down the differential equation whose general solution is y = AJ, /3(22) + BJ_1;3(22), with A, B arbitrary constants. (If you don’t already know the answer, try making the substit t = 22.) d) Identify the singular points of the equation whether it is regular or irregular. DO NOT SOLVE ution below, and specify for each singular point x(x —5)?y" + Any! + (x? — 25)y = 0. 9. Write the following system in matrix form, and find its general solution: dz/dt 382 + dy + 22 dy/dt -y dz/dt -2r + Ww ~ Qz 10. Find the solution of the following initial-value system of differential equations: So(2 Ge x @)-4(4).