[m4r35n357]

[Aside on Taylor Series] The section on recurrence relations is the basis of a little-known ODE solving technique sometimes known as the Taylor Series Method (occasionally Differential Transform). It allows an "extension" of Euler's Method to arbitrary order (using arbitrary precision arithmetic), by _calculating_ the higher derivatives rather than approximating them as in RK4 and friends.

Here is an open access link to just one of many papers on the subject: https://projecteuclid.org/euclid.em/1120145574

It includes a link to the actual software they used.

[EDIT] Obviously applying the technique above to dx/dt = x reproduces the Taylor Series in the article!