CARLETON UNIVERSITY
FINAL EXAMINATION December 2008 DURATION: 3 HOURS Department Name and Course Number: Mathematics and Statistics, MATH 1005F Course Instructor: Dr. A. Alaca PART I: Multiple Choice Questions. No partial marks. Circle the correct answer.
MATH 1005A Test 3 Solutions
March 9, 2010 [Marks] [3] Questions 1-5 are multiple choice. Circle the correct answer. Only the answer will be marked. 1. The sequence frn g1 converges for n=0 (a) jrj < 1 Solution: (e) [3] 2. lim [ln(n)]2 = n!1 n (b) 1 (c) 1
MATH 1005A Test 2 Solutions
February 23, 2010 [Marks] [3] Questions 1-5 are multiple choice. Circle the correct answer. Only the answer will be marked. 1. Two independent solutions of 2y 00 + 5y 0 3y = 0 are given by p p 5+ 37 5 37 x (a) y1 = ex=2 ; y2 =
MATH 1005A Test 1 Solutions
February 2, 2010 [Marks] [5] 1. Solve the initial-value problem 2y 0 = cos(x) ; y(0) = 2. y Solution: p 2yy 0 = cos(x) ) y 2 = sin(x) + c ) y = sin(x) + c: p p y(0) = 2 ) 2 = c ) c = 4 ) y = sin(x) + 4. x2 + y 2 ; y(1) = 2. xy
MATH 1005A - Notes 5 Systems of Equations
A system of two, linear, homogeneous equations with constant coecients has the form x0 = ax + by y 0 = cx + dy; where a; b; c and d are constants, and x and y are functions of t. The system may be expressed in mat
MATH 1005A - Notes 4 Second-Order Equations Reducible to the First Order
A second-order equation (linear or nonlinear) in which the dependent variable (e.g., y) does not appear explicitly can be reduced to a rst-order equation for y 0 . Thus, given the se
MATH 1005A - Notes 3 Cauchy-Euler Equations
An equation of the form x2 y 00 + Axy 0 + By = 0; where A and B are constants (or, equivalently, ax2 y 00 + bxy 0 + cy = 0, with a; b and c constants), is called a Cauchy-Euler equation. In standard form, the eq
MATH 1005A - Notes 2 Variation of Parameters
Consider a linear, nonhomogeneous equation in standard form, y 00 + p(x)y 0 + q(x)y = f (x); and let y1 and y2 be two, independent solutions of the associated homogeneous equation y 00 + p(x)y 0 + q(x)y = 0. We
MATH 1005A - Notes 1 Reduction of Order
Consider a linear, second-order, homogeneous equation in standard form, y 00 + p(x)y 0 + q(x)y 0 = 0: Suppose that one solution y1 is known. Then a second, independent solution y2 is obtained by letting y2 (x) = u(x
MATH 1005A Test 4 Solutions
March 23, 2010 [Marks] Questions 1 and 2 are multiple choice. Circle the correct answer. Only the answer will be marked. 1. Which of the following series converge(s)? (i) (a) All (b) (i) and (ii) (c) (i) and (iii)
1 X 3 2n n=0