Initial condition

Parameter in differential equations and dynamical systems In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time t = 0 {\displaystyle t=0} ) of a differential equation, difference equation, or other "time"-dependent equation which evolves in time. The most fundamental case, an ordinary differential equation of order k (the number of derivatives in the equation), generally requires k initial conditions to trace the equation's evolution through time. In other contexts, the term may refer to an initial value of a recurrence relation, discrete dynamical system, hyperbolic partial differential equation, or even a seed value of a pseudorandom number generator, at "time zero", enough such that the overall system can be evolved in "time", which may be discrete or continuous. The problem of determining a system's evolution from initial conditions is referred to as an initial value problem. Linear system Discrete time A linear matrix difference equation of the homogeneous (having no constant term) form X t + 1 = A X t {\displaystyle X_{t+1}=AX_{t}} has closed form solution X t = A t X 0 {\displaystyle X_{t}=A^{t}X_{0}} predicated on the