PREVIEW
4.2: REDUCTION OF ORDER
The technique reduction of order" and how to use it to construct a
second linearly independent solution given one solution to a linear
2nd order equation 1,1.
4.3: HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
PREVIEW
4.2: REDUCTION OF ORDER
The technique reduction of order" and how to use it to construct a
second linearly independent solution given one solution to a linear
2nd order equation 1,1.
4.3: HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
PREVIEW
5.1: LINEAR MODELS: INITIAL-VALUE PROBLEMS
Setting up the model for a mass on a spring, and the form of the various forces that affect its motion;
Understand the distinction between damped and undamped oscillations (both in terms of the equatio
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
LaplaceTransforms
Laplace transforms are used for solving linear differential equations by converting integral and
differential equations into algebraic equations which are gene
PREVIEW
6.3: SPECIAL FUNCTIONS
Basic understanding of power series from Calculus II, and how to work
with such series (e.g. differentiation, shifting index of summation) (this
is from 6.1);
General idea of finding a series solution to a linear 2nd orde
PREVIEW
1.3: DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS
Understand the basic concept of mathematical modeling, and the steps
involved;
Set up simple ODEs (with appropriate initial conditions) as mathematical models for a variety of physical phenomen
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
Solving Higher Order Ordinary Differential Equations
We will review solving higher order linear with constant coefficients, reduction of order and
variation of parameters.
Line
PREVIEW
4.4: UNDETERMINED COEFFICIENTS. SUPERPOSITION APPROACH (covered
on Assign #2 but not in lecture . . . will only be tested on this section in
short answer or multiple choice questions)
The form of the particular solution to be assumed.
4.6: VARI
PREVIEW
5.2: LINEAR MODELS: BOUNDARY VALUE PROBLEMS
Solve problems involving beams satisfying various boundary conditions such as embedded, free, and/or simply supported 3;
Solve various applications of 2nd order and higher boundary value
problems. In
PREVIEW
5.1: LINEAR MODELS: INITIAL-VALUE PROBLEMS
Setting up the model for a mass on a spring, and the form of the various forces that affect its motion;
Understand the distinction between damped and undamped oscillations (both in terms of the equatio
PREVIEW
5.2: LINEAR MODELS: BOUNDARY VALUE PROBLEMS
Solve problems involving beams satisfying various boundary conditions such as embedded, free, and/or simply supported 3;
Solve various applications of 2nd order and higher boundary value
problems. In
PREVIEW
4.1: LINEAR DIFFERENTIAL EQUATIONS: BASIC THEORY
What it means for a 2nd order equation to be linear; what it means to
be homogeneous 4.3;
Familiarity with operator notation 4.4, and how to use this to represent
differential equations 4;
Findi
PREVIEW
12.1: SEPARABLE PARTIAL DIFFERENTIAL EQUATIONS
Familiarity with the vocabulary of partial differential equations: linear vs nonlinear, order of an equation, homogeneous vs nonhomogeneous;
Understand what it means to solve a partial differential
PREVIEW
8.2: HOMOGENEOUS LINEAR SYSTEMS Contd
Concept of multiplicity for eigenvalues;
How to obtain a solution when an eigenvalue has multiplicity greater
than 1, but it generates sufficiently many independent eigenvectors;
In the case that the multi
PREVIEW
8.2: HOMOGENEOUS LINEAR SYSTEMS
How to solve a homogeneous linear system with constant coefficients
in the case that the eigenvalues are complex conjugates, while still
obtaining real-valued solutions;
Interpret the graphical behavior of the so
PREVIEW
7.1: DEFINITION OF THE LAPLACE TRANSFORM
Familiarity with piecewise functions and improper integrals from Calculus I and II;
Basic concept of an integral transform (specifically Laplace transform)
and its use in solving differential equations;
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
Solving 1st Order Ordinary Differential Equations
We will only consider three: linear, separable and exact.
Linear 1st Order Equations
Put the equation in the standard form
Academic Success Centre
academicsuccess@uoit.ca
http:/academicsuccess.uoit.ca
Solving Homogeneous Linear Systems using
Eigenvalues and Eigenvectors
What are the eigenvalues and eigenvectors?
If A is an
matrix, then a nonzero vector K is an eigenvector
PREVIEW
8.2: HOMOGENEOUS LINEAR SYSTEMS
Re-writing a higher-order differential equation as a system of 1st-order
differential equations (examples 1,2,3).
Finding eigenvalues and eigenvectors for a matrix 4;
Concept of multiplicity for eigenvalues;
Ho
PREVIEW
8.1: PRELIMINARY THEORY
How systems of differential equations arise in modelling various physical phenomena;
Writing a linear system in matrix notation 1;
The classification associated with systems of equations (linear vs nonlinear, homogeneou
MATH 2860U Differential Equations for Engineers
Midterm
Instructor: Isaac Ye
Thurs. July. 24, 2014
Last name:
First name:
Signature:
Student Number:
TA (circle one): Marian K; Camelia Y
Instructions
Please make sure your TA; otherwise, there will be 0.5
MATH 2860U Differential Equations for Engineers
Midterm
Instructor: M. Beligan, P. Grover
Mon. October. 22, 2012
Last name:
First name:
Signature:
Student Number:
TA (circle one): David B.; Jamil J.; Marian K.; Ruth (Qing Hong) Li.
Instructions
Before st
MATH 2860 Differential Equations for Engineers
MIDTERM
Instructor: A. Shakoori, M. Beligan
November, 2016
First Name
Last Name
Student Signature
Student Number
Name of your TA
(circle correct one)
Nicholas F.
Mia M.
Daniel P.
FIRST 4 LETTERS IN YOUR LAST
MATH 2860F2016
Midterm examination
1. Answer the following questions in the space provided.
Page 2 of 8
(a) [4 marks] A 1(kg) mass is attached to a spring with constant k = 4(N/m). There
is no damping, no external force. The mass is released from rest, wi
MATH 2860F2015
Midterm examination
1. Answer the following questions in the space provided.
Page 2 of 7
(a) [4 marks] Solve the following IVP: y00 4y0 + 4 = 0, y(0) = 1, y0 (0) = 3
ANS:
(b) [4 marks] Use the integrating factor method to solve x2 y0 + ( x
MATH 2860F2015
Midterm examination
1. Answer the following questions in the space provided.
Page 2 of 7
(a) [4 marks] Solve the following IVP: y00 4y0 + 4 = 0, y(0) = 1, y0 (0) = 3
ANS:
(b) [4 marks] Use the integrating factor method to solve x2 y0 + ( x