Euler's method

We have seen that the error in approximating the solution to an initial value problem is proportional to $\Delta t$. That is, if $E_{\Delta t}$ is the Euler's method approximation to the solution to an initial value problem at $\overline{t}$, then for some constant of proportionality $K$.

In this problem, we will see how to use this fact to improve our estimates, using an idea called accelerated convergence.

1. We will create a new appro…

In this problem, we'll modify Euler's method to obtain better approximations to solutions of initial value problems. This method is called the Improved Euler's method.

In Euler's method, we walk across an interval of width $\Delta t$ using the slope obtained from the differential equation at the left endpoint of the interval. Of course, the slope of the solution will most likely change over this interval. W…