MATH 1005A Test 2 Solutions
February 13, 2009 [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 =
z 2.2 Homogeneous Equation
A first order equation is homogeneous if it can be placed in the form
y
y0 = f ( )
x
y0 =
y 0 y2
x2 + y 2
, y = 2, y =
are homogeneous.
x
x
xy
y
A homogeneous equation is solved by making the substitution u = , where u is a new
Chapter 5
A set of n first-order differential equations which involve n unknown functions of an
independent variable t, where n 2, is called a system of first-order equations. A
system is linear if it has the form
x01 = a11 (t)x1 + a12 (t)x2 + + a1n (t)x
Note: all definitions and examples are from your textbook
CHAPTER 1
z 1.1 Basic Concepts
An ordinary differential equation of order n 1 is a relation which contains the
derivative of order n of an unknown function y (dependent variable), and may also con
z 3.3 Linear Nonhomogeneous Equations
Theorem: The general solution of the nonhomogeneous equation
a(x)y 00 + b(x)y 0 + c(x)y = g(x)
is y = yp + yh , where yp is any particular solution, and yh = c1 y1 + c2 y2 is the general solution
of the association h
z 2.4 Functions of Two Variables
A function f of two variables assigns to any pair (x, y) in its domain a real number denoted
by f (x, y), where the domain of f is either specified or taken to be all points (x, y) in the
plane where f (x, y) is defined.
MATH1005 Test 4 2:353:25, Nov. 25 2016
Total: 20 marks
Closed book, no GRAPHING calculator!
Question 1. [3] What can you say about the following series?
X
(1)n n
n+1
n=0
a) It converges conditionally (*)
It converges to the value 1
b)
It converges absolut
MATH1005H Solution-Test 1 - 14:3515:25, Sep. 30 2016
Total: 20 marks
Closed book, no GRAPHING calculator!
Question 1. [2] The equation
(a) Separable
dy
dx
=
x+y
xy
can be expressed as an equation of the type
(b) Homogeneous (*)
(c) Linear
(d) None of thes
MATH1005H Solution-Test 2 - 14:3515:25, Oct. 14 2016
Total: 20 marks
Closed book, no GRAPHING calculator!
Question 1. [3] The general solution of y 00 + 2y 0 + 10y = 0, x > 0 is given by
(a) y = c1 ex + c2 e3x
(b) y = ex [c1 cos(3x) + c2 sin(3x)]
(c) y =
MATH1005H Solution-Test 3 - 14:3515:25, Nov. 4 2016
Total: 20 marks
Question 1. [5] Solve the differential equation:
y 00 2y 0 + y = ex , x > 0.
Solution:
The associated homogeneous is y 00 2y 0 +y = 0, the general solution is erx where r2 2r +1 =
(r 1)2
CARLETON UNIVERSITY
FINAL EXAMINATION
April 2015
DURATION: 3 HOURS
SCANTRON FORMS REQUIRED
Department Name and Course Number: School of Mathematics and Statistics, MATH 1005
A, B, C, D, E, F
Course Instructor(s): Dr. S. Melkonian (Section A), Dr. B. Fodde
MATH 1005B Test 1 Solutions
January 28, 2008 [Marks] [5] 1. Solve the initial-value problem 2y 0 = sin(x) ; y (0) = 1. y
[5]
2. Find the general solution of y 0 + y =
Solution: p The equation is separable. 2yy 0 = sin(x) ) y 2 = cos(x) + c ) y = c cos(x):
MATH 1005A Test 4 Solutions
March 20, 2009 [Marks] [3] Questions 1-5 are multiple choice. Circle the correct answer. Only the answer will be marked.
1 X (1)n1 p 1. The series n+1 n=1
(a) Converges absolutely Solution: (b) [3] 2. The series
1 X (1)n n3 n=0
MATH 1005A Test 3 Solutions
March 6, 2009 [Marks] [3] Questions 1-5 are multiple choice. Circle the correct answer. Only the answer will be marked. 1. lim [ln(n)]2 = n!1 n (b) 1 (c) 1 (d) 1 2 (e) Does not exist
(a) 0
Solution: (a) [3] 2. The sequence frn
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.
CARLETON UNIVERSITY
FINAL EXAMINATION APRIL 2007 DURATION: 3 HOURS Department Name and Course Number: Mathematics and Statistics, MATH 1005 Course Instructor(s): Dr. A. Alaca, Dr. S. Melkonian, Dr. R. Krechetnikov
AUTHORIZED MEMORANDA
Non-programmable, no
MATH1005 Review
First-order Dierential Equations
1. Separable DE:
dy
dx
= f (x)g(y), dy/g(y) = f (x)dx.
2. Homogeneous eqn:
dy
dx
= g( xy ). Let v = xy . Then it is separable.
dy
3. 1st order linear eqn: dx
+P (x)y = Q(x). The solution is y(x)I(x) =
C, wh