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GILBERT STRANG: OK.

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So this is the second
lecture about these pictures,

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in the phase plane
that's with axes y and y

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prime, for a second order
constant coefficient

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linear, good problem.

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Good problem.

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And you remember that
we study that equation

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by looking for special
solutions y equals e to the st.

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When we plug that
into the equation,

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we get this simple
quadratic equation.

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And everything depends on that.

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So today this video
is about the case

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when the roots are complex.

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You remember, so the
roots, complex roots, you

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have a real part, plus or
minus an imaginary part.

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And this happens when b
squared is smaller than 4ac.

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Because you remember,
there's a square root

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in the formula for the solution
of a quadratic equation.

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There's a square root
of b squared minus 4ac,

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the usual formula from school.

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And if b squared is smaller,
we have a negative number

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under the square root.

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And we get complex roots.

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So last time the
roots were real.

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The pictures in the phase
plane set off to infinity,

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or came in to 0, more or less
almost on straight lines.

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Now we're going to have
curves and spirals, because

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of the complex part.

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So here are the three
possibilities now.

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We had three last time.

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Here are the other three
with complex roots.

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So the complex, the
real part, everything

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depends on this real
part that the stability

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going in, going out,
staying on a circle

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depends on that real part.

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If the real part is
positive, then we go out.

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We have an exponential e
to the a plus i omega t.

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And if a is positive that e to
the at would blow up, unstable.

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So that's unstable.

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Here is a center.

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When a is zero, then we just
have e to the i omega t.

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That's the nicest example.

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I do that one first.

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So in that case we're just
going around in a circle

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or around in an ellipse.

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And finally, the
physical problem

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where we have damping,
but not too much damping.

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So the roots are still complex.

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But they're going in.

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Because if a is negative,
e to the at is going to 0.

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So that's a stable case.

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That's a physical case.

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We hope to have a little damping
in our system, and be stable.

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This one we could
say neutrally stable.

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This one is certainly unstable.

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Let me start with that,
the neutrally stable.

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Because that's the most famous
equation in second order

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equation in mechanics.

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It's pure oscillation, a
spring going up and down,

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an LC circuit going back
and forth, pure oscillation.

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And we see the solutions.

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So I've written-- I've taken
this particular equation.

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You notice no damping.

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There's no y prime term.

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OK.

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So here is the solution,
famous, famous solution.

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And y prime it
will be c1, I guess

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the derivative of the
cosine is minus omega times

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sine omega t, plus c2.

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The derivative of that
is omega cos omega t.

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So that's the y and y prime.

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So for every t, it's going
to be an easy figure.

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Here is y.

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Here is y prime.

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And that's the phase
plane, phase plane.

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So at each time t, I
have a y and a y prime.

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And it gives me a point.

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So let me put it in there.

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As time moves on
that point moves.

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And it's the picture
in the phase plane,

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the orbit sometimes
you could say,

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it's kind of like
a planet or a moon.

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So for that, what is
the orbit for that one?

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Well it goes around
in an ellipse.

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It would be a circle
with-- let me draw it.

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This is the case omega equal 1.

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In that case, in that
most famous case,

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we simply go around a circle.

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There's y.

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There's y prime.

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We have cosine and sine and
cos squared plus sine squared

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is 1 squared.

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And we're going around a circle
of radius 1, or another circle

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depending on the
initial condition.

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Here there's a factor
omega, giving an extra push

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to y prime.

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So if omega was 2, for
example, then we'd have a 2

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in y prime from the omega,
which is not in the y.

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And that would make y
prime a little larger.

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And it would be twice as--
it would go up to twice as--

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that's meant to be, meant to
be an ellipse with height 2 up

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there, or in general
omega, and 1 there.

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So in the y direction
there is no factor omega.

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And we just have
cosine and sine.

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And that would be a typical
picture in the phase plane.

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But if we started with
smaller initial conditions,

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we would travel on
another ellipse.

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But the point is--
and these are called,

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this picture is called a center.

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So that's one of the six
possibilities, and in some way,

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kind of the most beautiful.

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You get ellipsis
in the phase plane.

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They close off.

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Because the solution just
repeats itself every period.

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It's periodic.

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y is periodic.

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y prime is periodic.

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They come around again
and again and again.

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No energy is lost, conservation
of energy, perfection.

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And I would say neutrally
stable, neutral stability.

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The solution doesn't go into 0.

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Because there's no damping.

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It doesn't go out to infinity.

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Because there's constant energy.

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And that's the picture
in the phase plane.

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OK.

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So that's the center.

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And now I'll draw one
with a source, or a sink.

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I just have to change
the sign on damping

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to get source or sink.

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So let me do that.

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So now I'm going to do
a spiral source or sink.

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This is the unstable one,
going out to infinity.

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This is the stable
one coming in to 0.

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And let me do y double prime,
plus or maybe minus 4y prime,

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plus 4y equals 0.

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Suppose I take that equation.

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Then I have s squared
plus 4s, oh maybe--

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maybe 2 is a nicer number.

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2 is nicer than 4.

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Let me change this
to a 2, and a 2.

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And so I have s
squared plus 2s plus 2

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or minus 2s plus 2 equals 0.

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So those are my-- positive
damping would be with a plus.

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So with a plus sign, the roots
are s squared plus 2s plus 2.

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The roots are 1, or rather
a minus 1 plus or minus i.

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Plus sign, and then the
minus sign, with a minus 2.

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Then all the roots have
a plus, plus or minus i.

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Everything is depending on
these roots, these exponents,

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which are the solutions of
the special characteristic

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equation, the simple
quadratic equation.

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And you see that depending
on positive damping

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or a negative damping, I get
stability or instability.

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And let me draw a picture.

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I don't if I can try two
pictures in the same thing,

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probably not.

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That wouldn't be smart.

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So what's happening then?

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Let's take this one.

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So this solution y
is e to the minus t.

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That's what's making it
stable coming into 0, times--

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and from here we have
c1 cos t and c2 sine t.

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That's what we get
from the usual,

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as in the case of a center that
carries us around the circle.

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So what's happening in this
picture, in this phase plane?

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Here's a phase plane
again, y and y prime.

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Without the minus
1, we have a center.

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We just go around in a circle.

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But now because of
the minus 1, which

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is the factor e to the
minus t in the solution,

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as we go around we come in.

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And the word for that
curve is a spiral.

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So this would be the center,
going around in a circle.

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But now suppose we start here.

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Suppose we start at y equal
1, and y prime equal 0,

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start there at time 0.

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Let time go.

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Plot where we go.

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Where does this y
and the y prime,

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where is the point, y, y prime?

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OK.

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I'm starting it at-- so I'm
probably taking c1 as 1,

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and c2 as 0.

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So I'm starting it right there.

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And then I'll travel
along, depending on sines.

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I would go, I think,
probably this way.

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So it will travel
on a-- it comes in

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pretty fast, of course.

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Because that exponential
is a powerful guy

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that e to the minus t.

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So this is the solution,
damping out to 0.

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That's with the plus
sign, plus damping,

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which gives the minus sign
in the s, in the exponent.

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And then so that
is a spiral sink.

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Sink meaning just as
water in a bathtub

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flows in, that's what happens.

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Now what happens in a
spiral source that's

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what we have with a minus sign.

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Now we have a 1.

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Now we have an e to the plus t.

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Everything is growing.

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So instead of decaying, we're
going around but growing-- OK.

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I'm off the board, way off
the board with that spiral.

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Which is going to
keep going around,

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but explode out to infinity.

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So those are the
three possibilities

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for complex roots, centers,
spiral source, and spiral sink.

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For real roots we
had ordinary source,

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and ordinary sink, no spiral.

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And then the other possibility
was a saddle point,

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where almost surely we go out.

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But there was one direction
that came into the saddle point.

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OK.

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Those six pictures
are going to control

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the whole problem of stability,
which is our next subject.

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Thank you.