A differential equation is an equation involving an unknown function and one or more of its derivatives. The equation is an ordinary differential equation (ODE) if the unknown function depends on only one independent variable. Some examples of ODEs follow:
In (1-5)
is the independent variable;
the dependent variables are 
, and 
, respectively.
In what follows we will frequently use the notation 
to represent
, 
to represent 
,
to represent 
, and, in general, 
to
represent 
.
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
having the following properties: 1) and its first 
derivatives
exist for all in , so and its first 
derivatives must
be continuous in , and 2) satisfies the differential equation
for all in .
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.
With a differential equation, we can associate initial conditions or boundary conditions, auxiliary conditions on the unknown function and its derivatives. If these conditions are specified at a single value of the independent variable, they are referred to as initial conditions and the combination of the differential equation and an appropriate number of initial conditions is called an initial value problem (IVP). If these conditions are specified at more than one value of the independent variable, they are referred to as boundary conditions and the combination of the differential equation and the boundary conditions is called a boundary value problem (BVP).
a) The logistic equation,
with initial condition 
;
for 
the solution is
b) The mass-spring system equation,
with the initial conditions 
, 
; for
, 
, 
, 
, 
,
,
the solution is
a) The differential equation,
with the boundary conditions
, 
; the solution
.
b) The differential equation,
with boundary conditions 
; the solution is
, for 
an arbitrary constant.
An th-order differential equation is said to be linear if it can
be written in the form
A nonlinear differential equation is
simply one that is not linear.
As examples, (4) is linear while (2), (3), and
(5) are nonlinear. Equation
(1) is linear when 
is a linear function of 
; otherwise, it is
nonlinear. Differential equations arising from first principle models are
generally nonlinear. Nonlinear equations do not usually yield to analytical
approaches and computational methods are called for.
Linear equations constitute a highly important class of differential equations in physics and engineering and are used in idealized models of such phenomena as mechanical vibrations, electrical circuits, planetary motions, etc. An important property of linear equations is that of superposition: To illustrate the superposition principle, consider the following IVP:
The data for this problem are
. If 
is
a solution with data 
, and 
is a solution with data 
, then the
principle states that 
is a solution
for the data 
. This idea extends readily to
th order differential equations. In practice, superposition permits us
to decompose a problem with complicated data into simpler parts, to solve
each problem separately, and then to combine these solutions to find the
solution to the original problem.
To solve the IVP,
we solve the following two problems:
a homogeneous equation with the original initial conditions, and
an inhomogeneous equation with zero initial conditions. The solution to the first problem is
and the second is
So, the solution to the original problem is the sum
A solution, , of a differential equation is said to be stable if
any other solution whose initial data is sufficiently close to that of
remains in a ``tube'' enclosing ; if the solution is not stable,
it is said to be unstable. If the diameter of the tube approaches
zero as becomes large, then is said to be asymptotically
stable.
In elementary treatments of differential equations it is assumed that the initial value problem has a unique solution that exists throughout the interval of interest and which can be obtained by analytical techniques. However, many of the differential equations encountered in practice cannot be solve explicitly, so we are led to methods for obtaining approximations to solutions. Such solutions are usually called numerical solutions. Matters are also complicated by the fact that solutions can fail to exist over the desired interval of interest. Even more troublesome are problems with more than one solution.
a) The differential equation,
does not have a solution that can be expressed in terms of elementary functions.
b) The IVP,
has the solution 
which exists on the interval 
but does not exist on the interval 
.
c) The IVP,
does not have a unique solution. In fact, it is not difficult to show that:
1) 
is a solution on any interval containing 
;
2) 
is a solution on any interval 
for any 
;
3) 
is a solution on 
and this is the largest such
interval on which 
is a solution.
