As noted, Euler's method is little used in practice, as there are much better ways of solving initial value problems. By better, we mean, "able to achieve a result of the same precision using a larger step size". [Euler's method is also unstable for some problems where the step size can take you outside the physical domain of the function and the solution runs away to infinity.]
To improve on Euler's method, we will use the fourth-order Runge-Kutta method. This method requires four evaluations of the differential at each step, but often allows a much larger step size to achieve the same result. The method can be summarised as:
where we have written h in place of the step size Δx.