Differential Equations
School of Mathematics and Statistics, Carleton University
(Carleton University)
Differential Equations
1 / 13
Outline
1
Second-order Linear Nonhomogeneous Equation(With Constant
Coefficient)
The method of Variation of Parameters
(Ca
Differential Equations
School of Mathematics and Statistics, Carleton University
(Carleton University)
Differential Equations
1 / 21
Outline
1
Second-order Linear Equation
Fundamental Theorem
Second-order Linear Homogeneous Equation(With Constant
Coeffici
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 15
Outline
1
Comparison Test / Limit Comparison Test
The Comparison Test for Divergence
The Limit Comparison Test for Divergence
Examples
E
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 28
Outline
1
The Integral Test and Estimates of Sums
The Integral Test for Divergence
Examples
Exercises
(Carleton University)
Infinite Ser
Differential Equations
School of Mathematics and Statistics, Carleton University
(Carleton University)
Differential Equations
1 / 29
Outline
1
Systems of Equations
Preliminary Theory
Eigenvalues and eigenvectors
Solutions to Homogeneous Linear Systems
(Ca
Orthogonal Functions and Fourier Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Orthogonal Functions and Fourier Series
1 / 57
Outline
1
Review
(Carleton University)
Orthogonal Functions and Fourier Series
2 / 57
Ou
Math 1005 A
Tutorial # 2
Date: June 14, 2012. In Tutorial
I
Tutorial Questions
Question I.1. The power-series representation of the function f (x) =
x4
about the point a = 0 is
1 x3
Question I.2. The coefficient of x7 in the Taylor series of the function
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 20
Outline
1
Strategy for Testing Series
Exercises
(Carleton University)
Infinite Series
2 / 20
List of Tests for
Test
P
n=1
an
Test Result
Math 1005 A
Sample Final
June 2013
Question 1. Multiple Choice Questions( Clearly enter your answers in the table provided
below. Only the answers written in this table will be graded).
Question
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Answer
Question
Ans
Math 1005 A, Summer 2013
Page 2 of4
concurrent registration in MATH 1104 (or MATH 1107], or permission of the School.
Tentative Schedulecfw_THIs SCHEDULE IS SUBJECT TO CHANGE DEPENDING ON THE PROGRESS
OF THE COURSE.)
WEEK SECTI
Differential Equations
School of Mathematics and Statistics, Carleton University
(Carleton University)
Differential Equations
1 / 18
Outline
1
Reduction of Order
(Carleton University)
Differential Equations
2 / 18
Outline
1
Reduction of Order
2
Cauchy-Eul
Differential Equations
School of Mathematics and Statistics, Carleton University
(Carleton University)
Differential Equations
1 / 24
Outline
1
Second-order Linear Nonhomogeneous Equation(With Constant
Coefficient)
Fundamental Theorem
The method of Undeter
MATH1004F Solution-Test 4 19:3520:25, Nov. 12 2014
Multiple Choice (No Partial Mark), circle the best possible answer
1. [2 points ] An antiderivative of f (x) = cot(4x + 5) is given by
(a)4 ln(sec(4x + 5) 1
(c)
1
4
(b)
ln | sin(4x + 5)| + 3 (*)
n
X
2. [2
MATH1004F Solution-Test 5 7:358:25, Nov. 19 2014
Total: 20 marks
Multiple Choice (No Partial Mark), circle the best possible answer
R
1. [2 points] Evaluate the indefinite integral (x 1)ex dx.
(b) xex + C
(a) The integral diverges
(c) xex + C (*)
(d) (x 1
Orthogonal Functions and Fourier Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Orthogonal Functions and Fourier Series
1 / 19
Outline
1
Properties of Sine and Cosine Functions
(Carleton University)
Orthogonal Funct
Taylor Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Taylor Series
1 / 15
Outline
1
Representations of Functions by Power Series
Representations of Functions: Substitution
Differentiation and Integration of Power S
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 24
Outline
1
Sequences
Definition
Convergence of Series
Exercises
(Carleton University)
Infinite Series
2 / 24
Sequences
Definition 1.1
A s
Taylor Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Taylor Series
1 / 28
Outline
1
Taylor And Maclaurin Series
Representations of Functions
Binomial Series
Application
Use series to evaluate the limit
Find the sum
Taylor Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Taylor Series
1 / 23
Outline
1
Power Series
Exercises
(Carleton University)
Taylor Series
2 / 23
Power Series
Definition 1.1 (Power Series)
A power series about
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 21
Outline
1
Absolute Convergence And The Ratio and Root Tests
Absolute Convergence
The Ratio Test
The Root Test
Rearrangements
Examples
Co
Math 1005 A
Test 2
May 30, 2013
Total mark: 30. Closed book. Non-programmable calculators are allowed.
Last Name
First Name
Student Number
Question 1. [4] Multiple Choice Questions(Please CIRCLE your answer )
(1) (2 points) The general solution of y 00 6y
Math 1005 A
Test 3
Jun 6, 2013
Total mark: 30. Closed book. Non-programmable calculators are allowed.
Last Name
First Name
Student Number
Question 1. [14 Points] Multiple Choice Questions(Please CIRCLE your answer )
(1) (2 points) The sum of the series
X
Infinite Series
School of Mathematics and Statistics, Carleton University
(Carleton University)
Infinite Series
1 / 19
Outline
1
Alternating Series
The Alternating Series Test for Divergence
The Alternating Series Estimating Theorem
Examples
Exercises
(Ca
School of Mathematics and Statistics. Carleton University Outline
ll Strategy for solving rst-unis?" equations
I Exercises
Dl'erenal Equations 9 54 m Strategy for solm'ng rst-order equations
1. First use algebra to put the equation in normal form y : f(x;