MATH 202 Exam 1, Fall 2002

1 MATHEMATICS 202 SECOND SEMESTER, 2004-05 QUIZ 1 Time: 55 MINUTES. Date: March 23, 2005. Name:————————————- ID Number:——————————– Circle Problem Session Instructor: Dr. A.Lyzzaik, Dr. H. Yamani Circle Section Number: 15, 16, 17, 18, 19 GRADE: 1. 2. 3. 4. 5. 6. —————————————————– Total: /50 2 1. Solve the initial-value problem dy dx = 1 + ey−x+5, y(6) = 1. (8 points) 3 2. Solve the differential equation x2 dy dx − 2xy = 3y4. (8 points) 4 3. Solve the initial value problem (x2 + y2 + xy) dx − xy dy = 0 y(1) = e − 1. (8 points) 5 4. Find the general solution of the differential equation 4x2y′′ + y = 0 for x > 0 if one of the solutions is y1 = x1/2 ln x. (8 points) 6 5. Find the general solution of the differential equation y′′′ + 6y′′ + y′ − 34y = 0. (8 points) 7 6. Answer TRUE (T) or FALSE (F) only: (a) ——- The set of functions {x2, x|x|} on ] − ∞, ∞[ is linearly indepen- dent. (2 points) (b) ——- An initial-value problem dy dx = f(x, y), y(x0) = y0 has a unique solution if f(x, y) and fx(x, y) are continuous in the xy−plane. (2 points) (c) ——- A set {y1, y2} of solutions of y′′ + y′ + y = 0 on ] − ∞, ∞[ is linearly independent if and only if the Wronskian W(y1, y2) is nonzero at some x ∈] − ∞, ∞[. (2 points) (d) ——- The initial-value problem (x2 + 1)y′′ + xy′ + y = 0 y(0) = y′(0) = 0 has the unique solution y = 0 on ] − ∞, ∞[. (2 points) (e) ——- The differential equation (cos x sin x − xy2) dx + y(1 − x2) = 0 is exact. (2 points)