Solving simple differential equations numerically

Differential equations arise in situations where a quantity evolves, usually over time, according to a given relationship. They are extremely common in engineering and physics, and appear quite naturally. One of the classic examples of a (very simple) differential equation is the law of cooling devised by Newton. The temperature of a body cools at a rate proportional to the current temperature. Mathematically, this means that we can write the derivative of the temperature of the body at time using the following differential equation:

Here, is a positive constant that determines the rate of cooling. This differential equation can be solved analytically by first separating the variables and then integrating and rearranging them. After performing this procedure, we obtain the general solution:

Here, is the initial temperature.

In this recipe, we will solve a simple ODE numerically...