Initial condition

Linear system

Discrete time

A linear matrix difference equation of the homogeneous (having no constant term) form ${\displaystyle X_{t+1}=A{X}_t}$ has closed form solution ${\displaystyle X_t=A^t{X}_0}$ predicated on the vector ${\displaystyle X_0}$ of initial conditions on the individual variables that are stacked into the vector; ${\displaystyle X_0}$ is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable X; that behavior is stable or unstable based on the eigenvalues of the matrix A but not based on the initial conditions.

Alternatively, a dynamic process in a single variable x having multiple time lags is

${\displaystyle x_t=a_1{x}_{t-1}+a_2{x}_{t-2}+\cdots +a_k{x}_{t-k}.}$

Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its characteristic equation ${\displaystyle {\lambda }^k-a_1{\lambda }^{k-1}-a_2{\lambda }^{k-2}-\cdots -a_{k-1}\lambda -a_k=0}$ to obtain the latter's k solutions, which are the characteristic values ${\displaystyle {\lambda }_1,\dots ,{\lambda }_k,}$ for use in the solution equation

${\displaystyle x_t=c_1{\lambda }_1^t+\cdots +c_k{\lambda }_k^t.}$

Here the constants ${\displaystyle c_1,\dots ,c_k}$ are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition ${\displaystyle x_t}$ Is known.

Continuous time

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

${\displaystyle \frac{dX}{dt}=AX.}$

Its behavior through time can be traced with a closed form solution conditional on an initial condition vector ${\displaystyle X_0}$. 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^th order linear equation in a single variable x is

${\displaystyle \frac{d^{k}x}{d{t}^k}+a_{k-1}\frac{d^{k-1}x}{d{t}^{k-1}}+\cdots +a_1\frac{dx}{dt}+a_{0}x=0.}$

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 ${\displaystyle {\lambda }^k+a_{k-1}{\lambda }^{k-1}+\cdots +a_1\lambda +a_0=0,}$ whose solutions are the characteristic values ${\displaystyle {\lambda }_1,\dots ,{\lambda }_k;}$ these are used in the solution equation

${\displaystyle x(t)=c_1{e}^{{\lambda }_{1}t}+\cdots +c_k{e}^{{\lambda }_{k}t}.}$

This equation and its first k – 1 derivatives form a system of k equations that can be solved for the k parameters ${\displaystyle c_1,\dots ,c_k,}$ given the known initial conditions on x and its k – 1 derivatives' values at some time t.