Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 1
Basic Concepts
A differential equation is any equation that includes derivatives, such as
or
dy
dt = y (1.1)
( ) 2
t 2 d2 y
dy
dt 2 + (1 − t) = e ty (1.2)
dt
There are two main classes of differential equations:
• ordinary differential equations (abbreviated ODES or DES) are
equations that contain only ordinary derivatives; and
• partial differential equations (abbreviated PDES) are equations
that contain partial derivatives, or combinations of partial and ordinary
derivatives.
In your studies you may come across terms for other types of differential
equations such as functional differential equations, delay equations,
differential-algebraic equations, and so forth. In order to understand any
of these more complicated types of equations (which we will not study this
semester) one needs to completely understand the properties of equations
that can be written in the form
dy
dt
= f(t, y) (1.3)
where f(t, y) is some function of two variables. We will focus exclusively
on equations of the form given by equation 1.3 and its generalizations to
equations with higher order derivatives and systems of equations.
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