This material is not a replacement for a course in differential equations, which courses tend to provide insights and methods that allow for algebraic solutions to many important differential equations, as well as providing insight into behavior of solutions that you can get without having to solve them in detail.
We provide it here because many traditional courses in differential equations ignore numerical computations entirely and we wish to show that these can be done with an amount of effort not much beyond what is involved in numerical integration, for all sorts of differential equations.
We will begin by solving a first order differential equation, then consider a second order equation, and finally one describing planetary motion, which is second order and has two dependent variables. (Though planets move in three dimensional space, their motions lie in a single plane. Our dependent variables are then the \(x\) and \(y\) coordinates of a planet and the independent variable is time \(t\).)
The major difference between these is in the number of columns that need be created.
What do you mean by "we". Are you going to do it while I go to sleep?
Well, I'll show you how to set one up, and you will see that you can change the equation without that much effort and solve them yourself which gives you powers unknown to previous generations of students.