MATH 202 Exam 1, Fall 2002

American University of Beirut MATH 202 Differential Equations Spring 2009 quiz # 1 Exercise 1 (15 points) Show that the IVP: dy dx = √ 1 − y2 , y(0) = 1 , has infinitely many solutions; does it contradict the existence and uniqueness theorem? justify. Exercise 2 a. (10 points) solve the differential equation: dy dx = y − y2. b. (5 points) are the solutions found in part a) all the solutions? justify. c. (5 points) give the solution of the IVP: dy dx = y − y2 , y(0) = −1; and give the interval of definition I. Exercise 3 (15 points) Solve the IVP: dy dx − 2 x y = x2 + 3 x , y(1) = 2 Exercise 4 (15 points) Use an appropriate substitution to solve the differential equation y′ y + ln y = √ 1 − ex Exercise 5 (15 points) Integrate g(x, y, z) = x √ y2 + 4 over the surface cut from the parabolic cylinder y2 + 4z = 16 by the planes x = 0, x = 1, and z = 0. Exercise 6 (20 points) Let C be the curve lying on the surface z = y2 + 1, whose sides lie vertically above the triangle in the xy-plane with x = 0, y = 0, x + y = 1 (see figure). 6 -  x x + y=1 z 1 1 z = y2+1 y Find � C 3xzdx + ydy + x2dz i. directly (you need to parameterize three curves on the surface!) ii. by using Stoke’s theorem