 Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be [and usually is] rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.The behavior of a system of ODEs can be visualized through the use of a phase portrait.Given a differential equation
a function , where "I" is an interval, is called a "solution" or integral curve for "F", if "u" is "n"-times differentiable on "I", and

Given two solutions and , "u" is called an "extension" of "v" if and
A solution that has no extension is called a "maximal solution".