MATH 202 Final, Summer 2011

Math 202 - Final (Summer 11) T. Tlas ================================================ • Please answer questions 4, 5 and 6 on the same sheet of paper on which they are written. Questions 1, 2 and 3 each have an extra sheet for you to write your answer on them. Any part of your answer written on the wrong page will not be graded. • There are 6 problems in total. Some questions have several parts. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : ================================================ Q1 Q2 Q3 Q4 Q5 Q6 TOTAL Problem 1 (4 points each) Solve the following IVPs: i- y′ = yx − x + y − 1 ; y(0) = 2 ii- y′′ − 3y′ + 2y = 6e−x ; y(0) = 0 ; y′(0) = −1 iii- y′′ + y = δ(t − 1) ; y(0) = 0 ; y′(0) = 0 iv- y′ + 6y + 9 � t 0 y(τ)dτ = δ(t − 1) ; y(0) = 0 ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (10 points) Solve the following BVP: 4xy′′ + 2y′ + y = 0 ; y(0) = 0 , y �π2 4 � = 1 ================================================ ADDITIONAL SHEET FOR PROBLEM 2 ANSWER Problem 3 (10 points) Solve the following IVP: ˙X =   2 −1 3 −2   X +   0 t   ; X(0) =   1 0   ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 (6 points) Prove the following identity involving the Bessel function: 2λJλ(x) = xJλ+1(x) + xJλ−1(x) where the series expression for the Bessel function is: Jλ(x) = ∞ � n=0 (−1)n n!Γ(1 + λ + n) �x 2 �2n+λ ================================================ Problem 5 (10 points) Let ⃗F be the vector field given by −−−−→ F(x, y) = − y x2 + y2 ⃗i + x x2 + y2 ⃗j Find � C ⃗F · d⃗r where C is the curve made of the straight line segment going from (1, 0) to (1, 1) followed by the straight line segment going from (1, 1) to (−1, 1), followed by the straight line segment going from (−1, 1) to (−1, 0). ================================================ Problem 6 (8 points) Let V be the following vector field: −−−−−−→ V (x, y, z) = ⃗k Let S be the upper unit hemisphere centered at the origin. Find the flux � S ⃗V · d ⃗A, where the normal vector is ‘outwards’ (away from the origin).