MATH 202 Exam 1, Spring 2002

MATHEMATICS 202 SPRING SEMESTER 2005-06 QUIZ I Time: 70 MINUTES. Date: March 11, 2006. Name:—————————————————————————- ID Number:———————————————————————– Section Number:—————————————————————— Course Instructors: Prof. Abdallah Lyzzaik and Dr. Hassan Yamani Question Grade 1 /14 2 /14 3 /14 4 /14 5 /14 6 /12 7 /18 TOTAL /100 Answer The Following Seven Questions On The Page Al- located For Each Question (You May Use The Back Of The Pages If Needed). 1 2 1. Solve the initial-value problem (14 points) dy dx = y2 − x2 yx , y(1) = − √ 2. 3 2. Solve the differential equation (14 points) (y3 + x + 1) dx + 3y2 dy = 0 by finding an appropriate integrating factor to make it exact. 4 3. Solve the initial-value problem (14 points) (3x − y2) dx − 4xy dy = 0, y(1) = 2. 5 4. Solve the differential equation (14 points) (x + y)2 dy dx = 1. 6 5. Find the general solution of the differential equation (14 points) x2y′′ − (x2 + 2x)y′ + (x + 2)y = 0 provided that y = x is a known solution. 7 6. Sketch the regions in the xy−plane for which the initial-value problem dy dx = � x2 − 1 1 − y2 , y(x0) = y0. possesses real and unique solutions; justify your answer. (12 points) 8 7. The differential equation x2y′′ − 4xy′ + 6y = 0 is known to have solutions y1 = x2 and y2 = x3. (a) Show that the Wronskian W(y1, y2) = 0 for every real x. (6 points) 9 (b) Show that the set {y1, y2} is linearly independent on (−∞, ∞). (6 points) (c) Do the results in (a) and (b) contradict? Justify your answer. (6 points)