The * order* of a differential equation is the order of the highest
derivative appearing in the equation.
Equations (2), (3), and (4) are
second order equations and (1) and (5)
are first order equations.

A * solution* of a general differential equation of the nth order,

is a real-valued function
defined over some interval
**I** having the following properties: 1) and its first **n** derivatives
exist for all **t** in **I**, so and its first **n-1** derivatives must
be continuous in **I**, and 2) satisfies the differential equation
for all **t** in **I**.

a) The function,

is a solution to the differential equation

b) The function

where and are arbitrary constants, is a solution to the differential equation

In this case, is also referred to as a * general*
solution
because all
solutions to the differential equation can be represented in this form for
appropriate choices of the constants and . The function
is a
* particular* solution because it contains no arbitrary constants.