MATH 202 Final, Summer 2010

Math 202 - Final (Summer 10) T. Tlas ================================================ • Please answer question 4 on the same sheet of paper on which it is written. Questions 1 and 2 have an extra sheet for you to write your answer on it, while question 3 has two extra sheets for this purpose. Any part of your answer written on the wrong page will not be graded. • There are 4 problems in total. Most questions have several parts. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : Section Number : ================================================ Q1 Q2 Q3 Q4 TOTAL Problem 1 (15 points each) Solve the following IVPs: i- y′ = yex ; y(0) = 1 ii- y′ = 1 Cos(x + 2y + 3) − 1 2 ; y(−3) = 0 iii- y′′ + y = f(t) ; y(0) = 0 ; y′(0) = 0 where f(t) =          0 0 ≤ t < π 1 π ≤ t < 2π 0 2π ≤ t iv- y′′′ − 3y′′ + 2y′ = 1 ; y(0) = 0 ; y′(0) = 1 2 ; y′′(0) = 2 ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (35 points) Find the 10th, 11th, 13th and the 100th coefficients of the series solution (centred at 0) of the following IVP (i.e. find the coefficients of x10, x11, x13, x100 of the power series solution of the IVP below) y′′ − 2 x y + 8 y = 0 ; y(0) = 1 y′(0) = 1 ================================================ ADDITIONAL SHEET FOR PROBLEM 2 ANSWER Problem 3 (20 points each) Solve the following IVPs: i- ˙X =      1 −1 2 −1 1 0 −1 0 1      X ; X(0) =      0 0 −1      ii- ˙x = y + δ(t − 1) ˙y = −x x(0) = 0 ; y(0) = 0 ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 (15 points) Is there a second-order, linear, ordinary differential equation such that both y1(x) = 1 and y2(x) = Sin(x) are solutions of it? In other words, can you find two functions P(x) and Q(x) such that the equation: y′′ + P(x)y′ + Q(x)y = 0? has y1(x) and y2(x) as its solutions? If yes, give an example. If no, provide a justifcation. ================================================