PREVIEW
3.1: LINEAR MODELS
Set up linear differential equations (with appropriate initial conditions)
as mathematical models for a variety of physical phenomena, and
solve the resulting equations;
Interpret solutions of differential equations in the co
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
7.4: OPERATIONAL PROPERTIES II
Understand and apply the relationship between multiplication by t
and the derivative of the Laplace transform 7.1;
Basic concept of a convolution of two functions 7.2, and how to apply
the convolution theorem 7.3;
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
5.2: 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
2.3: LINEAR EQUATIONS
Solving a linear equation by introducing an appropriate integrating
factor 1.
2.4: EXACT EQUATIONS
Recognizing if a first-order differential equation is exact 2.2;
Solving exact differential equations 2;
To find (where
rsLuE Saw no: 5
MATH 2860U Midterm examination Page 2 of 6
1. (a) [3 marks] Find the integrating factor associated with the DE (but DO NOT solve
the DE!)
(x + njZ + (x + 2)y = 22w"
B FM -\L -x _ _\_\ c X+\*\ c. J
1 31mm 32: 4 ZEMl _. 1: so 96) x x\-\
Academic Success Centre
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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 (same as on previous notes)
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 (
PREVIEW
7.4: OPERATIONAL PROPERTIES II
Finding the Laplace transform of a periodic function 7.1.
7.5: THE DIRAC DELTA FUNCTION
Basic concept of a unit impulse function and the Dirac delta function
7.2;
Finding the Laplace transform of the Dirac delta
MATH 2860U
Midterm examination
Page 2 of 6
1. (a) [3 marks] Find the integrating factor associated with the DE (but DO NOT solve
the DE!)
dy
( x + 1) + ( x + 2)y = 2xex
dx
ANS:
(b) [3 marks] Determine the general solution of the following linear homogeneo
PREVIEW
7.2: INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES
Finding the Laplace transform for a given initial value problem;
Solving the resulting algebraic equation once a differential equation
has had the Laplace transform applied;
Overall, solvin
PREVIEW (same as on previous notes)
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 (
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
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
1.1: DEFINITIONS AND TERMINOLOGY
Basic concept of a differential equation 1.1;
Familiarity with the vocabulary of differential equations: ODE 1.2 vs
PDE 1.3, linear vs nonlinear 1.4, order of an equation, systems of equations, etc.;
Ability to
MATH 2860S12 Assignment #2
DUE DATE: This assignment is to be submitted entirely on
paper in your TA's assignment drop box by FRI July 20 at
noon. Hand in one perfect copy per group of 2.
Last Name
First Name
1
2
Learning Objectives: This assignment allow
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
[email protected]
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
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