Let us now turn to a typical numerical problem, solving differential equations. We'll start with Ordinary Differential Equations (ODE). Namely, we consider equations of the type:
$$\tag{1} y'(t)=f(t,y(t))$$
The most famous algorithm is Euler's method and the most popular the RK4 variant of the Runge–Kutta methods. Methods can be explicit (the solution only depends on quantities already known) or implicit (the solution involves other unknowns which requires solving intermediate equations). The backward Euler method is an implicit method. Euler's method is the simplest one and the typical starting point to introduce the basic idea. Not surprisingly, it is also one of the least efficient one. We discretize time into steps $t_1$, $t_2$, $\cdots$, $t_n$ separated by the same interval $h$. The solutions at these times are denoted $y_n=y(t_n)$. Back to $(1)$, approximating $y'(t)\approx (y(t+h)-y(t))/h$, we have:
$$y_{n+1}=y_n+hf(t_n,y_n)$$