MATH 202 Midterm, Summer 2010

Math 202 - Midterm (Summer 10) T. Tlas ================================================ • Please answer question 4 on the same sheet of paper on which it is written. Questions 2,3 and 4 have extra sheets for you to write your answers on them. Any part of your answers written on the wrong page will not be graded. • There are 4 problems in total. Some questions have several parts to them. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : ================================================ Q1 Q2 Q3 Q4 TOTAL Problem 1 (11 points each) Solve the following IVPs: i- y′ = ex−y ; y(0) = 0 ii- y′ = −y x + 2ex2 ; y(1) = 1 iii- y′ = −2y + 3x + 2y2 x + 2xy ; y(0) = 0 ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (24 points) Solve the IVP y′′ − 2xy′ − 4y = 0 ; y(0) = 0 , y′(0) = 1 Also, find y(1). Hint: One of the series that you’ll obtain cannot be written in closed form, but this won’t be needed anywhere in this question. ================================================ ADDITIONAL SHEET FOR PROBLEM 2 ANSWER Problem 3 (11 points each) Solve the following IVPs: i- x2y′′ + xy′ + y = 0 ; y(1) = 1 , y′(1) = 0 ii- y′′ − 4y′ + 4y = ex ; y(0) = 0 , y′(0) = 0 iii- x2y′′ + (y′)2 = 0 ; y(1) = −1 , y′(1) = −1 ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 (10 points) Suppose you have two functions y1(x) and y2(x) which satisfy the following two equations: y′ 1 = y2 y′ 2 = −y1 If you are also given that y1(0) = 1, y′ 1(0) = 0, find y1(x) and y2(x). ================================================