## 1.1 Some Elements of the Theory     continued...

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.