where denotes the approximation to computed by the forward Euler method. Note that the ``driving function'' is evaluated at time , not . As a result, given, and the input vector for all , Eq.(7.10) can be iterated forward in time to compute for all . Since is an arbitrary function, we have a solver that is applicable to nonlinear, time-varying ODEs Eq.(7.8).
Because each iteration of the forward Euler method depends only on past quantities, it is termed an explicit method. In the LTI case, an explicit method corresponds to a causal digital filter . Methods that depend on current and/or future solution samples (i.e., for ) are called implicit methods. When a nonlinear numerical-integration method is implicit, each step forward in time typically uses some number of iterations of Newton's Method (see §7.4.5 below).
Backward Euler Method
General Nonlinear ODE