information:extraction
Continuous time

A differential equation system of the first order with n variables stacked in a vector X is

Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X 0 {\displaystyle X_{0}}. The number of required initial pieces of information is the dimension n of the system times the order k = 1 of the system, or n. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.

A single k order linear equation in a single variable x is

Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In this case the k initial pieces of information will typically not be different values of the variable x at different points in time, but rather the values of x and its first k – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The characteristic equation of this dynamic equation is λ k + a k − 1 λ k − 1 + ⋯ + a 1 λ + a 0 = 0, {\displaystyle \lambda ^{k}+a_{k-1}\lambda ^{k-1}+\cdots +a_{1}\lambda +a_{0}=0,} whose solutions are the characteristic values λ 1, …, λ k; {\displaystyle \lambda _{1},\dots,\lambda _{k};} these are used in the solution equation

This equation and its first k – 1 derivatives form a system of k equations that can be solved for the k parameters c 1, …, c k, {\displaystyle c_{1},\dots,c_{k},} given the known initial conditions on x and its k – 1 derivatives' values at some time t.