GILBERT STRANG: OK. Well, the idea of
this first video is to tell you what's coming,
to give a kind of outline of what is reasonable to learn
about ordinary differential equations. And a big part of
the series will be videos on first order
equations and videos on second order equations. Those are the ones you
see most in applications. And those are the ones you
can understand and solve, when you're fortunate. So first order equations
means first derivatives come into the equation. So that's a nice equation
that we will solve, we'll spend a lot of time on. The derivative is-- that's
the rate of change of y-- the changes in the unknown
y-- as time goes forward are partly from depending
on the solution itself. That's the idea of a
differential equation, that it connects the changes
with the function y as it is. And then you have
inputs called q of t, which produce their own change. They go into the system. They become part of y. And they grow, decay,
oscillate, whatever y of t does. So that is a linear equation
with a right-hand side, with an input, a forcing term. And here is a
nonlinear equation. The derivative of y. The slope depends on y. So it's a differential equation. But f of y could be y squared
over y cubed or the sine of y or the exponential of y. So it could be not linear. Linear means that
we see y by itself. Here we won't. Well, we'll come
pretty close to getting a solution, because it's
a first order equation. And the most general first
order equation, the function would depend on t and y. The input would
change with time. Here, the input depends only
on the current value of y. I might think of y
as money in a bank, growing, decaying, oscillating. Or I might think of y as
the distance on a spring. Lots of applications coming. OK. So those are first
order equations. And second order have
second derivatives. The second derivative
is the acceleration. It tells you about the
bending of the curve. If I have a graph, the
first derivative we know gives the slope of the graph. Is it going up? Is it going down? Is it a maximum? The second derivative tells
you the bending of the graph. How it goes away
from a straight line. So and that's acceleration. So Newton's law-- the
physics we all live with-- would be acceleration
is some force. And there is a force that
depends, again, linearly-- that's a keyword-- on y. Just y to the first power. And here is a little bit
more general equation. In Newton's law,
the acceleration is multiplied by the mass. So this includes a physical
constant here, the mass. Then there could
be some damping. If I have motion, there may
be friction slowing it down. That depends on the first
derivative, the velocity. And then there could be the
same kind of forced term that depends on y itself. And there could be some outside
force, some person or machine that's creating movement. An external forcing term. So that's a big equation. And let me just
say, at this point, we let things be nonlinear. And we had a pretty good chance. If we get these
to be non-linear, the chance at second
order has dropped. And the further
we go, the more we need linearity and maybe
even constant coefficients. m, b, and k. So that's really
the problem that we can solve as we get good at
it is a linear equation-- second order, let's say--
with constant coefficients. But that's pretty
much pushing what we can hope to do
explicitly and really understand the
solution, because so linear with constant
coefficients. Say it again. That's the good equations. And I think of
solutions in two ways. If I have a really nice
function like a exponential. Exponentials are
the great functions of differential equations, the
great functions in this series. You'll see them over and over. Exponentials. Say f of t equals-- e to the t. Or e to the omega t. Or e to the i omega t. That i is the square
root of minus 1. In those cases, we will get
a similarly nice function for the solution. Those are the best. We get a function that we
know like exponentials. And we get solutions
that we know. Second best are we get some
function we don't especially know. In that case, the
solution probably involves an integral of
f, or two integrals of f. We have a formula for it. That formula includes
an integration that we would have to
do, either look it up or do it numerically. And then when we get to
completely non-linear functions, or we have
varying coefficients, then we're going
to go numerically. So really, the wide,
wide part of the subject ends up as numerical solutions. But you've got a
whole bunch of videos coming that have nice
functions and nice solutions. OK. So that's first order
and second order. Now there's more, because
a system doesn't usually consist of just a single
resistor or a single spring. In reality, we have
many equations. And we need to deal with those. So y is now a vector. y1, y2, to yn. n different unknowns. n different equations. That's n equation. So here that is
an n by n matrix. So it's first order. Constant coefficient. So we'll be able
to get somewhere. But it's a system of
n coupled equations. And so is this one with
a second derivative. Second derivative
of the solution. But again, y1 to yn. And we have a matrix,
usually a symmetric matrix there, we hope, multiplying y. So again, linear. Constant coefficients. But several equations at once. And that will bring in
the idea of eigenvalues and eigenvectors. Eigenvalues and eigenvectors
is a key bit of linear algebra that makes these
problems simple, because it turns
this coupled problem into n uncoupled problems. n first order equations that
we can solve separately. Or n second order equations
that we can solve separately. That's the goal with
matrices is to uncouple them. OK. And then really the big
reality of this subject is that solutions
are found numerically and very efficiently. And there's a lot to learn
about that, a lot to learn. And MATLAB is a
first-class package that gives you numerical
solutions with many options. One of the options
may be the favorite. ODE for ordinary
differential equations 4 5. And that is numbers 4, 5. Well, Cleve Moler, who
wrote the package MATLAB, is going to create a
series of parallel videos explaining the steps
toward numerical solution. Those steps begin with
a very simple method. Maybe I'll put the
creator's name down. Euler. So you can know that because
Euler was centuries ago, he didn't have a computer. But he had a simple
way of approximating. So Euler might be ODE 1. And now we've left Euler behind. Euler is fine, but not
sufficiently accurate. ODE 45, that 4 and 5 indicate a
much higher accuracy, much more flexibility in that package. So starting with
Euler, Cleve Moler will explain several
steps that reach a really workhorse package. So that's a parallel series
where you'll see the codes. This will be a
chalk and blackboard series, where I'll find
solutions in exponential form. And if I can, I would like to
conclude the series by reaching partial differential equations. So I'll just write some partial
differential equations here, so you know what they mean. And that's a goal
which I hope to reach. So one partial
differential equation would be du dt-- you see
partial derivatives-- is second derivative. So I have two variables now. Time, which I always have. And here is x in
the space direction. That's called the heat equation. That's a very important
constant coefficient, partial differential equation. So PDE, as distinct from ODE. And so I write down one more. The second derivative of u
is the same right-hand side second derivative
in the x direction. That would be called
the wave equation. So this is like the first
order equation in time. It's like a big system. In fact, it's like an infinite
size system of equations. First order in time. Or second order in time. Heat equation. Wave equation. And I would like to also
include a the Laplace equation. Well, if we get there. So those are goals for
the end of the series that go beyond some courses in ODEs. But the main goal
here is to give you the standard clear picture
of the basic differential equations that we can
solve and understand. Well, I hope it goes well. Thanks.
