In this sub unit, I'll present a particular example of a differential equation
that we'll solve and analyze in a number of different ways.
Differential equations 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 we'll finish this lesson tomorrow morning.
The differential equation I mentioned last night
is known as Newton's Law of Cooling,
and here is what this equation says in words.
It says that the rate of cooling or warming, either way,
is some function of the temperature, T,
the temperature of the object that's 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,
it's 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.