Editing Initial condition (section)

{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Short description|Parameter in differential equations and dynamical systems}}{{Multiple image
| direction = vertical
| image1 = Nonsmooth initial condition for vibrating string.svg
| caption1 = An initial condition of a vibrating string
| image2 = Vibrating string oscillation for nonsmooth initial condition.gif
| caption2 = Evolution from the initial condition
| alt1 = A nonsmooth initial condition for a vibrating string, and the evolution thereof
}}

In [[mathematics]] and particularly in [[dynamic systems]], an '''initial condition''', in some contexts called a '''seed value''',<ref>{{cite book |last=Baumol |first=William J. |authorlink=William Baumol |title=Economic Dynamics: An Introduction |url=https://archive.org/details/economicdynamics0000baum_c7i2 |url-access=registration |location=London |publisher=Collier-Macmillan |edition=3rd |year=1970 |isbn=0-02-306660-1 }}</ref>{{rp|pp. 160}} is a value of an evolving [[Variable (mathematics)|variable]] at some point in time designated as the initial time (typically denoted ''t''&nbsp;=&nbsp;0). For a system of [[order (differential equation)|order]] ''k'' (the number of time lags in [[discrete time]], or the order of the largest derivative in [[continuous time]]) and [[dimension (vector space)|dimension]] ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional [[coordinate vector]]), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. 

In both [[differential equation]]s in continuous time and [[difference equation]]s in discrete time, initial conditions affect the value of the dynamic variables ([[state variable]]s) at any future time. In continuous time, the problem of finding a [[closed form solution]] for the state variables as a function of time and of the initial conditions is called the [[initial value problem]]. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.