We're almost at the end of this unit, and differential equations.
Differential equations are a vast topic,
- entire books have been written about the subject,
- big ,large, thick books - so what we've done in this unit
is just the beginning - it's just scratched the surface.
Nevertheless, I hope we've covered enough about differential equations
and how to think about them
so that you can view them as dynamical systems
and then we'll be able to study their properties
in the next several units.
I'll end this unit by first talking a little bit about
some notation and terminology that I haven't gotten into yet,
and then I'll summarize the main ideas
about differential equations that are presented in this unit.
After that, I'll say a little bit about Newton, and Laplace,
and determinism, and this will set the stage for the next unit
where will encounter chaos, and the butterfly effect.
So far we've looked at equations of this form:
dX/dT - some function of X,
but there are other types of differential equations,
and some vocabulary and terminology
associated with these different types,
that I think we should go over.
So, if the right hand side of the equation does not depend on time
- that's the case here - the derivative is a function just of X,
- how fast the cold beer is warming up,
is a function only of the temperature of the beer.
For such a situation, we say that the equation is autonomous.
In a sense it's just doing its own thing, it doesn't depend on time,
but if there is time dependence here,
then one would say
the right hand side of the equation is non-autonomous.
That might arise if, say, the room temperature in which the beer
or hot coffee was, was varying what time
- it warmed up in the day, and cooled off at night,
then, in order to figure out how fast something's cooling off,
we would need to know not only its current temperature,
but also the time day.
In any event, we will be studying only autonomous equations in this course.
There are lots of fun and interesting non-autonomous equations,
but we'll get plenty of mileage out of autonomous equations.
Another key term about differential equations is
the idea of the order of a differential equation.
The order of a differential equation is just
the highest derivative that appears in the equation.
Here, this is just the first derivative of X,
and so we would say that this is first order.
If there were first and second derivatives around
then we would say it's second order.
The second derivative - it's not important for this course,
but as you might guess - it's the derivative of the derivative,
- the rate change of the rate of change.
In this course we'll study only first-order equations.
At first this might seem like a limitation,
because there's some very important second order equations:
Newton's law of motion - perhaps the most important for mechanics and chaos,
is a second order equation,
however, it's possible to convert a second order equation
into a system of two first order equations.
That's not immediately obvious,
and we're not going to actually do that in this course,
- we could talk about in the forums if people are interested,
but the main point is that we'll only study first order equations
but that's not limiting us in any way.
OK, a little bit more terminology:
a differential equation is called ordinary
if it contains only ordinary i.e. full or total derivatives.
So, ordinary differential equations - that's what we'll talk about here
is often written ODE, and in contrast,
a non ordinary differential equation is not a peculiar differential equation
but it's called a partial differential equation,
and it involves partial derivatives - things like: this:
- this is a version of the wave equation.
We won't cover partial differential equations in this course,
we'll study only ordinary differential equations.
Next, I want to say a little bit about
the important ideas of existence and uniqueness.
So, let me state this result,
and then I'll talk about its implications.
So, consider a differential equation of this kind,
- that's what we've been studying in this unit,
and the initial condition is given,
so, we know the rule, we know the starting point.
If this function of X is a nice function,
- by that I mean: its continuous, it doesn't have any jumps,
and it's smooth, it doesn't have any sudden kinks or bends in it,
then, the solution to this equation exists and is unique.
So, let's think about what this means, and why it matters.
First of all, - these conditions:
that this function is smooth and continuous,
they're met in most physical applications and in most modelling applications.
It's very rare that these conditions aren't met.
So, what that means is, is that if I asked you a question like this:
here's a differential equation, I tell you the starting point,
- that there's one, and only one, solution,
So, that if I find a solution, and you find a solution.
they're the same solution - there's one, and only one out there,
and I hope that this result isn't surprising.
I've talked throughout this unit
about how this differential equation is a rule,
- it's a rule that tells how the rate of change of X,
is related to X, and so what this result says is that
the rule specified by this equation,
and the starting point, is unambiguous - there is only one solution to it,
there's only one way to follow the rule - it's an unambiguous rule.
So, in brief, this result,
which is proved in most differential equation's text books,
says that: differential equations of this sort are well-behaved.
If we have a reasonable right hand side, we're guaranteed to have one,
and only one, solution - this rule is unambiguous.