Backward Euler Method

An example of an implicit method is the backward Euler method:

$$\underline{\hat{x}}(n) \isdefs \underline{\hat{x}}(n-1) + T\dot{\underline{\hat{x}}}(n) \eqsp \underline{\hat{x}}(n-1) + Tf[n,\underline{\hat{x}}(n),\underline{u}(n)] \protect$$ (8.12)

Because the derivative is now evaluated at time $n$ instead of $n-1$ , the backward Euler method is implicit. Notice, however, that if time were reversed, it would become explicit; in other words, backward Euler is implicit in forward time and explicit in reverse time.