MATH 202 Final, Spring 2005

fawrtuan t NIVERSET | LIBKARY FINAL EXAMINATION, MATH 202 et JUNE 8, 2005; 3:00-5:00 P.M. Name: Signature: Number: Section: 13, 14, 15, 16 1. Instructions: e No calculators are allowed. e There are two types of questions: PART I consists of seven subjec- tive questions, and PART II consists of five multiple-choice questions of which each has exactly one correct answer. e Give detailed solutions for the problems of PART I in the provided space and circle the appropriate answers for the problems of PART II. 2. Grading policy: e 10 points for each of the problems of PART I. e 5 points for each correct answer of PART II. ¢ 0 point for no answer or more than one answer of PART II. GRADE OF PART I: GRADE OF PART II: TOTAL GRADE: Part I(1). Find the general solution of the system X’ = AX where 7 -2 4 0 -2 0 -1 (2). Find the general solution of the differential equation y" +y = 2sine + 5e. . Part (3). Use Laplace transform to solve the initial value problem . Part I(4). Find the general solution of the differential equation mame El de + | tl ay =o Very 2 Very 2, YO” . Part 1(5). Show that y = /z J.(z) is a solution for the differential equation . Part I(6). Solve the differential equation vy tir y=y?. . Part I(7). Find the general solution of the differential equation zy" ~ sry. +y=4alnz, «>0. . Part II: Multiple Choice Problems. Part II(1). The solution of the initial-value problem ytl=e sing, y(0) =0, passes through the point (a) (x = 7/2,y = 7/2 —1n2). (b) (@ = 1/2,y = —7/2 — In 2). (c) (c = 24/2,y = 7/24 1n2). (d) (2 = 1/2,y = —7/24+1n2). (e) None of the above. Part II(2). Knowing that the differential equation 227y" ~- 3ry' + (8+a)y =0 has a regular singular point at x = 0 with an indicial root r, then an application of Frobenius method to find the associated series solution Ro Cnt” yields for the coefficients cn, n > 1 the recurrence relation (a) Cn = Cn-i/[(n +r + 1)(2n + 2r — 3)]. (b) Cn = —Cp-i/[(n + r — 1)(2n + 2r — 3)). (c) Cp = ~Cn-1/[(n + 7 — 1)(2n + 2r + 3)). (d) er = Cra/[(m + 7 — 1)(2n + 2r — 3), (e) None of the above. 10 . Part II(3). Answer TRUE (T) or (FALSE) only: (a) —~ If m is a positive integer and J, is Bessel’s function of the first kind of order m, then J_m = {-1)"Jm. (b) —- Bessel’s function of the first kind of order v is defined by = ray (-1)" (@y"" Aa nt PC ni Tw +n) 2 , (c) —— The Gamma function is defined by T(z) = [ ett? | dt, 2 >0. Jo {d) —- Picard’s theorem guarantees a unique solution for the initial value problem dy Ge FE ¥), y(Zo) = Yo o if f and f, are continuous functions in some open rectangle containing (Xo, Yo): (e) —- The convolution of functions f and g on (0, 00) is defined by t (Fa gtt) = [ f@glr +8) ar Part II(4). (a) 6(t + 2) + (eb — e*)/t. (b) 6(t + 2) — (e' — e*)/t. (c) d(t — 2) + (ef —e*)/t. (d) 6(t — 2) — (et -—e*)/t. (e) None of the above. . Part II(5). The solution of the initial-value problem y(t) =cost + [lr t)cos(t—7)dr, y(0) =1, satisfies (a) y(2) =5 (b) y(2) = 2 (c) y(2) =3 (4) y(2) =4 (e) None of the above. Part II(6). The indicial roots of the differential equation are Qay" — (1+ 227) — zy =0 (a) 1,3/2. (b) 0,3/2. (c) -1, 2/3. (d) 0, -3/2. (e) None of the above. 11