MATH 202 Final, Spring 2002

FINAL EXAMINATION, MATH 202 June 8, 2005; 3:00-5:00 P.M. Name: Signature: Number: Section: 13, 14, 15, 16 1. Instructions: • No calculators are allowed. • 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. • 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: • 10 points for each of the problems of PART I. • 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: 1 2 Part I(1). Find the general solution of the system X′ = AX where A =    7 −2 4 −4 0 −2 0 0 −1    . 3 (2). Find the general solution of the differential equation y′′ + y = 2 sin x + 5e2x. 4 Part I(3). Use Laplace transform to solve the initial value problem t y′′ − y′ = t2, y(0) = 0. 5 Part I(4). Find the general solution of the differential equation � x √x2 + y2 − y x2 � dx + � y √x2 + y2 + 1 x � dy = 0. 6 Part I(5). Show that y = √x Jν(x) is a solution for the differential equation x2 y′′ + � x2 − ν2 + 1 4 � y = 0. 7 Part I(6). Solve the differential equation x2 y′ + 2x y = y3. 8 Part I(7). Find the general solution of the differential equation x2y′′ − xy′ + y = 4x ln x, x > 0. 9 Part II: Multiple Choice Problems. Part II(1). The solution of the initial-value problem y′(t) = cos t + � t 0 y(τ) cos(t − τ)dτ, y(0) = 1, satisfies (a) y(2) = 2. (b) y(2) = 3. (c) y(2) = 4. (d) y(2) = 5. (e) None of the above. Part II(2). L−1 � e−2s + ln �s − 1 s + 1 �� . (a) δ(t − 2) + (et − e−t)/t. (b) δ(t − 2) − (et − e−t)/t. (c) δ(t + 2) + (et − e−t)/t. (d) δ(t + 2) − (et − e−t)/t. (e) None of the above. Part II(3). The indicial roots of the differential equation 2xy′′ − (1 + 2x2)y′ − xy = 0 are (a) −1, 3/2. (b) 0, −3/2. (c) 1, 2/3. (d) 0, 3/2. (e) None of the above. 10 Part II(4). Knowing that the differential equation 2x2y′′ − 3xy′ + (3 + x)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 �∞ n=0 cnxn+r yields for the coefficients cn, n ≥ 1 the recurrence relation (a) cn = cn−1/[(n + r − 1)(2n + 2r − 3)]. (b) cn = −cn−1/[(n + r − 1)(2n + 2r + 3)]. (c) cn = −cn−1/[(n + r − 1)(2n + 2r − 3)]. (d) cn = cn−1/[(n + r + 1)(2n + 2r − 3)]. (e) None of the above. Part II(5). The solution of the initial-value problem y′ + 1 = e−(x+y) sin x, y(0) = 0, passes through the point (a) (x = π/2, y = −π/2 + ln 2). (b) (x = π/2, y = π/2 + ln 2). (c) (x = π/2, y = −π/2 − ln 2). (d) (x = π/2, y = π/2 − ln 2). (e) None of the above. 11 Part II(6). Answer TRUE (T) or (FALSE) only: (a) —– The Gamma function is defined by Γ(x) = � ∞ 0 e−ttx−1 dt, x > 0. (b) —- The convolution of functions f and g on [0, ∞) is defined by (f ∗ g)(t) = � t 0 f(t)g(τ + t) dτ. (c) —- Bessel’s function of the first kind of order ν is defined by Jν(x) = ∞ � n=0 (−1)n n! Γ(ν + n) �x 2 �2n+ν . (d) —- If m is a positive integer and Jm is Bessel’s function of the first kind of order m, then J−m = (−1)mJm. (e) —- Picard’s theorem guarantees a unique solution for the initial value problem dy dx = f(x, y), y(x0) = y0 if f and fx are continuous functions in some open rectangle containing (x0, y0).