University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
1.1: DEFINITIONS AND TERMINOLOGY
Basic concept of a dierential equation 1.1;
Familiarity with the vocabulary of dierential equations: ODE 1.2 vs
PDE 1.3, linear vs nonlinear 1.4, order of an equation, systems of equations, etc.;
Ability to rec
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 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, solvi
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 8.2: HOMOGENEOUS LINEAR SYSTEMS
Rewritin g a higherorder differential equation as a system of lstorder
differential equations (examples.
Finding eigenvalues and eigenvectors for a matrixEp
Concept of multiplicity for eigenvalues;
How to sol
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 12.1: SEPARABLE PARTIAL DIFFERENTIAL EQUATIONS
Familiarity with the vocabulary of partial differential equations: lin
ear vs nonlinear, order of an equation, homogeneous vs nonhomoge
neouslll;
Understand what it means to solve a partial differ
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 8.2: HOMOGENEOUS LINEAR SYSTEMS Contd
Concept of multiplicity for eigenvalues 8.2
How to obtain a solution when an eigenvalue has multiplicity greater
than 1, but it generates sufciently many independent eigenvectors
[323E
In the case that th
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 7.3: OPERATIONAL PROPERTIESI
Applying the shift properties to help find transforms and inverse trans
formsm. and to apply this when solving odesg
Understand the appearance of 1the unit step function in finding inverse
Laplace transforms [I];
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
. 8.1: PRELIMINARY THEORY
How systems of differential equations arise in modelling various phys
ical phenomena;
Writing a linear system in matrix notation Ill
The classication associated with systems of equations (linear vs non
linear, homoge
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 7.1: DEFINITION OF THE LAPLACE TRANSFORM
Familiarity with piecewise functions and improper integrals from Cal
culus  and II;
Basic concept of an integral transform (specically Laplace transform)
and its use in solving differential equations;
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 8.2: HOMOGENEOUS LINEAR SYSTEMS
How to solve a homogeneous linear system with constant coefcients
in the case that the eigenvalues are complex conjugates, while still
obtaining realvalued solutions;
Interpret the graphical behavior of the solu
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 6.4: 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 nding a series solution to a linear 2nd order
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
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
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
1.2: INITIALVALUE PROBLEMS (IVPs)
Understand how to nd the solution of an IVP based on the solution
of an ode and given initial conditions 2;
The conditions under which a unique solution is known to exist for a
rstorder equation 1.1;
How to
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
2.2: SEPARABLE EQUATIONS
What it means for an ODE to be separable 2.1, and recognizing if a
given dierential equation is separable or not;
How to solve a separable rst order ODE 1.
2.3: LINEAR EQUATIONS
How to recognize linear rstorder diere
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 2.4: EXACT EQUATIONS
— Recognizing if a ﬁrst—order differential equation is exact 2.2;
— Solving exact differential equations 1;
— To ﬁnd (where possible) an integrating factor which makes a ﬁrst—order
ode exact 2, and to subsequently solve the
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 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;
— F
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
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
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 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.
o 4.6: VARIAT
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
5.1: LINEAR MODELS: INITIALVALUE PROBLEMS
Setting up the model for a mass on a spring, and the form of the various forces that aect its motion;
Understand the distinction between damped and undamped oscillations (both in terms of the equations
University of Ontario Institute of Technology (UOIT)
Differential Equations for Engineers
ENGINEERIN Math2860

Fall 2015
PREVIEW
o 7.4: OPERATIONAL PROPERTIES ll
The Laplace transform of an integral, and how to solve integrodiffer
ential equationsl
Finding the Laplace transform of a periodic function IE].
0 7.5: THE DIRAC DELTA FUNCTION
Basic concept of a unit impulse fu