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GILBERT STRANG: OK.
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This is the second
video for Chapter 3.
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And it's going to
be pictures again.
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But it's pictures for a
second order equation.
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And I'll make them--
these will be nice.
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We'll know formulas here.
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These will be
constant coefficient,
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linear second order equations.
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And we know that
the solution-- there
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are two special
solutions, e to the s1 t
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and e to the s2t,
two null solutions,
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and any combination
is a null solution.
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So we're talking
about null equations,
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0 on the right-hand side.
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And we just want to
draw that picture that
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goes with solutions like that.
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So here is the magic word,
phase plane, phase plane.
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We're going to draw the
pictures in a plane.
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Because that's what
a blackboard is.
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And the axes we'll choose
will be y and y prime, not t.
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You'll see how t, time,
comes into the picture.
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But we have the two axes
will be y and y prime.
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So I had to figure
out what y prime was.
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It just brings down
an s1 from that term,
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and brings down an
s2 from that term.
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And now here's the example.
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Here is the first example.
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So I took this
particular equation.
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Notice that the damping
term is negative.
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I have negative damping.
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This will be unstable.
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Solutions will go
out to infinity.
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And I can find those solutions.
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Because you know that
I look for e to the st.
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I always look for e to the
st. I plug in e to the st.
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I get an s squared
from two derivatives,
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minus 3s from one derivative,
plus 2 equaling 0.
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I factor that, and I find
the 2s1 is 1, and s2 is 2.
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And now I'm ready for
the phase plane picture.
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OK.
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Phase plane picture, so
here are my solutions.
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s is 1 or 2.
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Then the derivative
has a 1 or a 2.
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And here's my plane.
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Here's my plane.
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And I want to draw on
that the solutions.
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These solutions, I
actually have formulas.
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I just want to draw them.
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So I'm plotting.
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One example would be that
c1 be 1, and let c2 be 0.
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So that's gone.
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c1 is 1.
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I just have that picture.
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What kind of a picture
do I have in the phase
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plane, in the y, y prime
plane, when that's y
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and that's y prime?
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Well, those are equal.
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So y equals y prime
for that solution.
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y equals y prime along
the 45-degree line.
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It's just like y equal x.
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y prime is y.
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And what's happening
on this 45-degree line?
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The solution is this solution,
is going straight out the line.
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As t increases, y and
y prime both increase.
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I go out.
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This is t going to infinity.
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And what about t going
to minus infinity?
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Because we got the
whole picture here.
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When t goes to minus infinity
that goes to 0, that goes to 0.
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Here is the point where
the universe began.
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The Big Bang is right there
at t equal minus infinity.
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And as t increases,
this point, y, y prime,
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is traveling along
that 45-degree line.
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Because y equals y
prime, and out there.
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And what about the
rest of the line?
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Well, if c1 was negative,
if c1 was negative
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I'd have a minus there,
and a minus there.
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I would just have minuses.
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And I'd be going out that line.
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Well, that's one line in my
whole plane, but not all.
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Now let me take as a
second line c1 equals 0.
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So nothing from e to the t, and
let me take y as e to the 2t,
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and y prime then
would be 2e to the 2t.
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OK.
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What's happening in the phase
plane for this solution,
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now looking at this one?
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Well in this solution, in this
case, y prime is 2 times y.
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y prime is 2 times y.
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So I'm staying on
the line y prime,
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where y prime is 2 times y.
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It's a steeper
line, steeper line.
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So that was the case, this
was the line where c2 was 0.
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There was no e to the 2t on
that first line that we drew.
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In the second line
that we drew, c1 is 0.
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There's no e to the t.
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Everything is in e to the 2t.
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So now c1 is 0 on this line.
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OK.
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And we just go out it.
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As t increases, y
prime increases faster.
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Because of the factor 2.
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So it goes up steeply.
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And it goes this way.
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When c2 is negative, if I
took a minus and a minus,
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I would just go down the
other way on the same line.
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And this is still the Big
Bang, t equal minus infinity,
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where everything starts.
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OK.
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So that is two lines,
the two special lines
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in the phase plane.
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But now I have to draw
all the other curves.
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And they will be curves.
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And where will they come from?
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They will come
from a combination.
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So now I'm ready for that one.
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Let me take the case
c1 equal 1, c2 equal 1.
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Yeah, why not? c1 equal 1.
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So I can erase c1.
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c2 equal 1, I can erase c2.
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And now I have another
solution, y and y prime.
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And I want to put it
in the phase plane.
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So at every value of t, at
every value of t that's a point.
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That's a value of y.
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This is a value of y prime.
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I plot the points y and y prime,
and I look at the picture.
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And again as t
changes, as t changes
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I'll travel along the solution
curve in the phase plane.
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I'll travel along.
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As t changes, y will
grow, y prime will grow.
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I'll head out here.
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But I won't be on that straight
line or that straight line.
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Because those were the
cases when I had only one
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of the two solutions.
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These were the
special solutions.
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And now I have a combination.
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So what happens as
t goes to infinity?
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As t goes to
infinity, this wins.
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As t goes to infinity,
the e to the 2t
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is bigger than e to the t.
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So this is the larger term.
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So it approaches.
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This curve now will
approach closer and closer
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to the one when the
line with slope 2.
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The 2 will be the
winner out here.
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But at t equal minus
infinity, near the Big Bang,
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at t equal minus infinity, e
to the 2t is even more small.
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So at t equal minus infinity,
or t equal minus 10,
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let's say, this is
e to the minus 10.
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This would be e to the minus
20; very, very, very small.
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These would win.
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So what happens
for this solution
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is it starts out along the line
given by the not-so-small t,
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the not-so-small exponent.
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It starts up that line.
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But t is increasing.
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When t passes some point,
this 2t will be bigger than t.
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And it will, I guess,
at t equals 0, 2t
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will be bigger than t.
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And from that point on, from
the t equal 0 point-- oh,
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I could even plot
the t equals 0.
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So at t equals 0, y
is 2 and y prime is 3.
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So at 1, 2, 1, 2, 3;
somewhere in there.
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So you see, the curve starts up
along the line where e to the t
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is bigger.
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They have the same size at
t equals 0, both equal 1.
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This is at t equals 0.
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And then for large
times, this one wins.
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So I approach that line.
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I don't know if you
can see that curve.
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And I don't swear to the
slopes of that curve.
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But in between in
there is filled
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with curves that start
out with this slope,
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and end with that slope.
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And the same here, it'll
start with this slope.
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But then go-- probably
this is a better picture.
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Yeah.
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That's a better picture.
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Yeah.
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It will just go up with slope.
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At the end it will have
slope 2 going upwards.
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Yeah.
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That looks good.
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Well, you could say I only
drew part of the phase plane.
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And you're completely right.
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If I start somewhere here, what
would you think would happen?
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What would you
think would happen
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if I start with that value of y
that much, and that value of y
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prime?
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It would have some mixture of--
there would be a c1 and a c2.
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So the other curves
that I haven't drawn yet
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come from the other c1 and c2.
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I've done c1 equal
1, and c2 equal 1.
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And c1 equal c2 equal 1.
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But now I have many
more possibilities.
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And what they do is they will--
so suppose I start there.
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It will approach--
this is the winner.
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This is the winner.
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Where c1 is 1, where
this is happening,
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there is the winner
for large time.
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So all curves swing up
toward parallel to that line.
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Or down here, they swing
down parallel to that line.
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So things here will
swing down this way.
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That's the phase plane.
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May I do one more example to
show that this was a source?
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This is called a source.
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Because the solution
goes to infinity.
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Wherever you start, the
solution goes to infinity.
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It's unstable, totally unstable.
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Now if I change to
a positive damping,
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then I would have
a plus sign there.
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These would be plus signs.
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I would have s equal minus
1, or s equal minus 2.
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So with positive damping,
I damp out naturally.
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And this picture
would be the same,
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except all the lines
are coming in to 0, 0.
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The solutions are damping to
0, 0; to nothing happening.
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So I just track the same lines,
but in the opposite direction.
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So instead of this
being the Big Bang,
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it's the end of the
universe, t equal infinity.
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OK.
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I'm up for one more picture
of this possibility.
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And let me take the equation
y double prime equal 4y.
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So my equation will be s squared
equal 4, s equals 2 or minus 2.
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And when I draw the phase
plane and the solutions,
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the solutions will be c1 e to
the 2t, and c2 e to the minus
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2t, from a 2 and a minus 2.
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That's the solution we all know.
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And now I should
compute its slope.
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y prime will be 2c1 e to the
2t minus 2c2 e to the 2t.
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And now you just want me
to draw those pictures.
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You just want me to
draw those pictures,
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and let me try to say
what happens here.
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This is a saddle point.
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It's called a saddle, when we
have in one direction things
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are growing, but in the
other, things are decreasing.
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So most solutions,
if c1 is not 0,
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then the growth is going to
win, and that will disappear.
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But there is the
possibility that c1 is 0.
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So there will be one
line coming from there.
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There will be one line
coming from there.
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Maybe I can try to draw that.
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Again, I'll draw that
pure line, where c1 is 0.
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So that pure line is coming.
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These are minuses here.
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So that line is coming
in to the center.
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So that's why we have a saddle.
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We approach a saddle if along
this where this is minus 2
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00:15:15,165 --> 00:15:17,950
of that, so I
think it would be--
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00:15:17,950 --> 00:15:21,310
so it's a slope of minus 2.
250
00:15:21,310 --> 00:15:29,050
So I think a slope like
that, so again this is y.
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00:15:29,050 --> 00:15:31,220
This is y prime.
252
00:15:31,220 --> 00:15:33,270
This is the slope of minus 2.
253
00:15:35,870 --> 00:15:37,780
And that's this curve.
254
00:15:42,750 --> 00:15:49,160
So it will be very exceptional
that we're right on that line.
255
00:15:49,160 --> 00:15:55,460
All other points, all other
curves in this phase plane,
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00:15:55,460 --> 00:15:58,180
are going to have a
little c1 in them.
257
00:15:58,180 --> 00:16:00,350
And then this will take over.
258
00:16:00,350 --> 00:16:05,250
And that gives us, as we
saw before, this slope.
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00:16:05,250 --> 00:16:07,180
This is 2 times that.
260
00:16:07,180 --> 00:16:12,360
So that line is where
everybody wants to go.
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00:16:12,360 --> 00:16:16,900
And only if you start
exactly on this line
262
00:16:16,900 --> 00:16:21,760
do you get this picture, and
you come into the saddle.
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00:16:21,760 --> 00:16:24,770
Instead of the Big Bang,
or the end of the universe,
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00:16:24,770 --> 00:16:27,910
this is now the
saddle point, where
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00:16:27,910 --> 00:16:32,630
we come in on this most
special of all lines,
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00:16:32,630 --> 00:16:34,960
coming from this picture.
267
00:16:34,960 --> 00:16:38,540
But almost always this
is the dominant thing.
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00:16:38,540 --> 00:16:40,120
And we go out.
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00:16:40,120 --> 00:16:43,170
So if I take a typical
starting point,
270
00:16:43,170 --> 00:16:52,280
I'll go out this like
that, or like this, oh no.
271
00:16:52,280 --> 00:16:53,170
Yeah, no.
272
00:16:53,170 --> 00:16:54,460
I'll go out.
273
00:16:54,460 --> 00:16:56,170
It'll have to go out.
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00:16:56,170 --> 00:16:59,720
So if I start anywhere
here, these are probably
275
00:16:59,720 --> 00:17:07,374
they're hyperbolas going
out in that direction.
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00:17:07,374 --> 00:17:09,750
I don't swear that
they're hyperbolas.
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00:17:09,750 --> 00:17:13,200
Here again we might start in.
278
00:17:13,200 --> 00:17:15,710
Because we have
big numbers here.
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00:17:15,710 --> 00:17:18,680
But then e to the t takes over.
280
00:17:18,680 --> 00:17:20,190
And we go out.
281
00:17:20,190 --> 00:17:21,980
So those go out.
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00:17:21,980 --> 00:17:25,670
These go out.
283
00:17:25,670 --> 00:17:29,420
And these go out.
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00:17:29,420 --> 00:17:32,540
So this is the big line.
285
00:17:32,540 --> 00:17:36,360
That's the line
coming from here.
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00:17:36,360 --> 00:17:39,130
And that's where
everything wants to go,
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00:17:39,130 --> 00:17:43,360
and everything eventually
goes that way, except the one
288
00:17:43,360 --> 00:17:45,100
line where c1 is 0.
289
00:17:45,100 --> 00:17:49,700
So this dominant term
is not even here then.
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00:17:49,700 --> 00:17:51,970
And then we should
become inwards.
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00:17:51,970 --> 00:17:56,280
So that saddle point
is the special point
292
00:17:56,280 --> 00:18:00,320
where you could go out,
if you go the right way.
293
00:18:00,320 --> 00:18:05,761
Or you could come in, if you go
the other special, special way.
294
00:18:05,761 --> 00:18:06,260
OK.
295
00:18:06,260 --> 00:18:08,370
So that sources,
sinks and saddles.
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00:18:08,370 --> 00:18:12,010
And I still have to
draw the pictures, which
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00:18:12,010 --> 00:18:17,080
involves spirals that
come from complex s,
298
00:18:17,080 --> 00:18:18,640
where we have oscillation.
299
00:18:18,640 --> 00:18:21,030
That'll be the next video.