## 8.1. Introduction

Any equation involving the derivatives, *y*¢, *y*², … , of a function *y* is
called a **differential equation**
(DE). Any function
y=f(
x
)
that satisfies the DE is called a **solution**. The **order**
of a DE is the order of the highest derivative in it. Examples:

(a)
y
′
=y
is 1^{st} order with solutions
y=c
e
x
,
where *c* is any constant.

(b)
y
′
′
+16y=0
is 2nd order with solutions
y=
c
1
cos4x+
c
2
sin4x
,
where *c*_{1} and *c*_{2} are arbitrary constants.

Experiences have shown that it is hopeless
to seek methods for solving *all* DE’s. Instead, it is more fruitful to ask whether a
given DE has any solution, and what properties of the solutions, if exist, can
be deduced from the DE. Thus, DEs provide
a new source of functions.

Results of great generality about the
solutions of DEs are usually difficult to obtain. However, there are a few exceptions, e.g.,
the **linear** DEs that are our only
concern.