Explicit forward Euler approximation

The explicit (or forward) Euler method begins by approximating the time derivative with a first-order finite difference as... 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

The forward Euler method is thus referred to as an explicit method, because Xk+i is taken to depend only on points that have been previously calculated, that is, x + depends only on The discretization used in the backward Euler method leads to the following approximation of the ODE... 

In contrast to the forward (explicit) Euler method, which uses the slope at the left-hand side to step across the interval, the imphcit verison of the Euler method crosses the interval by using the slope at the right-hand side, as shown in Figure 6.8. The implicit formula does not give any direct approximation ofy +i, instead an iterative method, e.g. the Newton method, is added inside the loop, thus advancing the differential equation to solve fory +1. This obviously comes at the price of more computation, but allows stability... 

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