In this sub unit, I'll present a
particular example of a differential
equation that we'll solve and analyze a
number of different ways.
The differential equation that I'll use is known as: Newton's Law of Cooling.
It describes how objects warm up or cool off.
You're probably familiar with it already even if you haven't studied physics.
For example, suppose you're home, after a
long hard day of recording videos about
derivatives and you open a cold beer....
So, I just took this beer out of my refrigerator, where, let's say, it's 5 degrees.
The room temperature is probably around 20 degrees.
So, as we probably have all experienced, this cold beverage is going to warm up,
and it will warm up very quickly at first.
The rate of warming up is largest when the difference
between the beer temperature and the room temperature is the largest.
So it will warm up very quickly at first, and then less quickly, and less quickly,
until it reaches room temperature at 20 degrees.
So, I'm aware that this beer is warming up, so I think I'm going to focus on this, and will finish this lesson tomorrow morning...
The differential equation I mentioned last
night is known as Newton's Law of Cooling,
and here's what this equation says in
words: the rate of cooling or warming,
either way, is some function of the
temperature T, the temperature of the object that is warming or cooling.
And, in particular, that function is proportional to the difference between the object's temperature and the room temperature.
So, if the object is a lot hotter, or a lot colder, than the room temperature, its temperature will change very quickly.
If the difference between the object's temperature and the room temperature
is very little, it won't be warming up or cooling off very quickly.
So, this is Newton's Law of Cooling in words. Let's take a look at the equation.