394
7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
If we are given a differential equation and an initial value for the unknown function,
we say that we have an initial value problem. For instance,
dv
dt
= 1.5 − 0.5v, v(0) = 0.5
is an initial value problem. In this situation, we know the value of v at one time and
we know how v is changing. Consequently, there should be exactly one function v that
satisfies the initial value problem.
This demonstrates the following important general property of initial value problems.
Initial value problems that are “well behaved” have exactly one solution, which exists
in some interval around the initial point.
We won’t worry about what “well behaved” means—it is a technical condition that will
be satisfied by all the differential equations we consider.
To close this section, we note that differential equations may be classified based on
certain characteristics they may possess. Indeed, you may see many different types of
differential equations in a later course in differential equations. For now, we would like to
introduce a few terms that are used to describe differential equations.
A first-order differential equation is one in which only the first derivative of the function
occurs. For this reason,
dv
dt
= 1.5 − 0.5v
is a first-order equation while
d 2 y
dt 2 = −10y
is a second-order equation.
A differential equation is autonomous if the independent variable does not appear in
the description of the derivative. For instance,
dv
dt
= 1.5 − 0.5v
is autonomous because the description of the derivative dv/dt does not depend on time.
The equation
dy
dt
= 1.5t − 0.5y,
however, is not autonomous.
7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
If we are given a differential equation and an initial value for the unknown function,
we say that we have an initial value problem. For instance,
dv
dt
= 1.5 − 0.5v, v(0) = 0.5
is an initial value problem. In this situation, we know the value of v at one time and
we know how v is changing. Consequently, there should be exactly one function v that
satisfies the initial value problem.
This demonstrates the following important general property of initial value problems.
Initial value problems that are “well behaved” have exactly one solution, which exists
in some interval around the initial point.
We won’t worry about what “well behaved” means—it is a technical condition that will
be satisfied by all the differential equations we consider.
To close this section, we note that differential equations may be classified based on
certain characteristics they may possess. Indeed, you may see many different types of
differential equations in a later course in differential equations. For now, we would like to
introduce a few terms that are used to describe differential equations.
A first-order differential equation is one in which only the first derivative of the function
occurs. For this reason,
dv
dt
= 1.5 − 0.5v
is a first-order equation while
d 2 y
dt 2 = −10y
is a second-order equation.
A differential equation is autonomous if the independent variable does not appear in
the description of the derivative. For instance,
dv
dt
= 1.5 − 0.5v
is autonomous because the description of the derivative dv/dt does not depend on time.
The equation
dy
dt
= 1.5t − 0.5y,
however, is not autonomous.