The general form of an ODE with order
is
As in the case of second-order ODEs, such an ODE can be classified as linear or nonlinear. The general form of a linear ODE of order
is
If
is the zero function, the equation is said to be
homogeneous. This discussion is primarily restricted to that case.
Many methods for solving linear second-order ODEs can be generalized to linear ODEs of order
, where
is greater than 2. If the order of the ODE is not important, it is simply called a linear ODE.
