Implicit Euler Method

Aside from the Explicit Euler method, we may also have an implicit relation where a dependent variable is not isolated in the equation; sometimes we can convert between the two, but this is not always possible.

In these cases, we can write the definition of a derivative slightly differently, replacing $h$ with $-h$. This creates a completely different equation, but one that is evaluated identically under the limit:

$$\frac{df(x)}{dx}=\underset{h\to 0}{\lim }\frac{f(x)-f(x-h)}{h}\approx \frac{f(x)-f(x-\Delta x)}{\Delta x}$$

This is again a finite difference, but a backward difference approximation (BDA) instead of a forward difference approximation (FDA).

For implicit relations, this can give rise to the implicit Euler method, or aptly named backward Euler method, similar to before (just shifting time forwards slightly to neaten the formula)…

$$\frac{y(t)-y(t-\Delta t)}{\Delta t}\approx g(t,y(t))⟹\boxed{y(t+\Delta t{)}_{\text{approx}}\approx y(t)+\Delta t\cdot g(t+\Delta t,y(t+\Delta t))}$$