An **n**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 **u**; 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
**n**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.