A linear ODE with
constant coefficients can be easily solved once the roots of the auxiliary equation (or characteristic equation) are known. Some examples of this type follow.
The characteristic equation of this ODE has real and distinct roots: 4, 1, and 7. Hence the solution is composed entirely of exponential functions.
The characteristic equation of this ODE has two pairs of equal roots:
and
. The repeated roots give rise to the
basis of the solutions
, 
.
The characteristic equation for this ODE has two pairs of roots with nonzero imaginary parts:
,
,
, and
. Hence the solution basis can be expressed with trigonometric and exponential functions.
