MATH 202 Exam 2, Fall 2012

Math 202 - Quiz 2 (Fall 12) T. Tlas ================================================ • Write the answer to question 4 on its sheet. The first three questions have extra sheets for you to write your answers on them. Any part of your answers written on the wrong sheet will not be graded. • There are 4 problems in total. The first question has two parts to it. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : Section : ================================================ Q1 Q2 Q3 Q4 TOTAL Problem 1 (15 points each) Solve the following IVPs: i- ˙X =         3 1 0 0 0 3 0 0 0 0 3 1 0 0 0 3         X ; X(0) =         1 0 1 2         ii- ˙X =   0 0 1 −1   X +   1 2   ; X(0) =   2 2   ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (30 points each) Solve the following IVP: ˙x = 2x − 2y + δ(t − 1) ; x(0) = 0 ˙y = 4x − 2y ; y(0) = 1 ================================================ ADDITIONAL SHEET FOR PROBLEM 2 ANSWER Problem 3 (30 points) Solve the following IVP: x2y′′ + (5x − x2)y′ + (3 + x)y = 0 ; lim x→0 � xy(x) � = 1 , lim x→0 � xy(x) �′ = −2 3 Additionally find y(2). Hint: Solve the equation using series centered at 0. ================================================ ADDITIONAL SHEET FOR PROBLEM 3 ANSWER Problem 4 (10 points) Suppose {an}∞ n=0 is a sequence which satisfies the following identity for all x: ∞ � n=0 (n + 1)an+1xn = − � ∞ � n=0 anxn �2� ∞ � n=0 xn n! � . Then, if a0 = 1 find a2012. ================================================