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MATH2860U:
Chapter 1
1
INTRODUCTION TO DIFFERENTIAL
EQUATIONS
Definitions and Terminology (Section 1.1, pg. 2)
Recall:
Back in Calculus II, we introduced the concept of a differential equation.
Definition:
A
differential equation
is an equation which contains the derivatives of an
unknown function (dependent variable) with respect to one or more independent
variables.
Notation:
In order for us to talk about differential equations and figure out how to solve them, it
helps to classify them according to a few criteria:
type, order, and linearity.
Ordinary Differential Equation vs Partial Differential Equation
Definition:
An
ordinary differential equation
(
ODE
) is one in which the unknown
function depends on a single
independent variable.
A
partial differential equation
(
PDE
) is one in which the unknown function depends on more than one independent
variables.
Order of a Differential Equation
Definition:
The
order
of a differential equation (either ODE or PDE) is the order of the
highest derivative in the equation.
1
Differential Equation

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Chapter 1
2
Note:
If we write a differential equation in the form
)
,
,
,
,
,
(
)
1
(
n
n
n
y
y
y
y
x
f
dx
y
d
where
f
is a real-valued continuous function, then the differential equation is said to be in
normal form
.
Linear vs Nonlinear Differential Equations
Definition:
An
n
th
-order differential equation
0
)
,
,
,
,
,
(
)
(
n
y
y
y
y
x
F
is said to be
linear
if
F
is linear in
)
(
,
,
,
,
n
y
y
y
y
.
i.e. any general linear ODE of order
n
is:
)
(
)
(
)
(
)
(
)
(
0
1
)
1
(
1
)
(
x
g
y
x
a
y
x
a
y
x
a
y
x
a
n
n
n
n
Any equation which is not of this form is called a
nonlinear
equation.
Note:
Nonlinear DEs are typically much harder to solve than linear DEs; because of this,
linearization
is a common strategy which is used to obtain a linear problem from a
nonlinear one in the hopes that the linear problem is easier to analyze.
Systems of Differential Equations
Another way to classify differential equations is based on the number of unknown
functions that are involved.
If a differential equation has more than one unknown
(dependent variable), then more than one function is to be determined and hence, more
than one equation is required…in this case, we have a
system
of differential equations.
2
Differential Equation