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GILBERT STRANG: OK.

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00:00:01,000 --> 00:00:05,490
So today is unforced--
that means zero

3
00:00:05,490 --> 00:00:09,830
on the right-hand side, looking
for null solutions-- damped--

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that means there
is a coefficient B

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00:00:12,560 --> 00:00:14,600
in the first derivative.

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And what's the solution?

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This is really a
basic, basic equation.

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00:00:20,790 --> 00:00:24,130
In many applications,
A would be the mass.

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00:00:24,130 --> 00:00:27,750
In a spring, for example,
A would be a mass.

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00:00:27,750 --> 00:00:30,250
B is the damping, the friction.

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And C is the spring
constant, the force

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00:00:34,290 --> 00:00:37,010
that pulls the mass back.

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Or in electronics, B
would be the resistance.

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It's giving some friction,
giving some heat.

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So that's our equation.

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Just we have to be
able to solve it.

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And we want to look
for exponentials.

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A pure exponential is just
right for a constant coefficient

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00:01:00,970 --> 00:01:02,420
equation like that.

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So I'll substitute y equals e
to the st. And what happens?

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Well, so there's a C e
to the st. That's a Cy.

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The derivative brings down an s.

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Two derivatives
bring down s squared.

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So I simply have As
squared, Bs, and C

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00:01:24,530 --> 00:01:31,760
all together, multiplying e
to the st. And how'd I get 0?

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Well the 0 is not going
to come from e to the st,

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so it has to come from this.

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So fundamentally, the
whole video and more

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is about a quadratic equation.

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As squared plus Bs
plus C equals 0.

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We have to solve it.

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We have to understand how
does the answer depend

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on these numbers, A and
B and C, the constants?

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OK?

35
00:02:02,160 --> 00:02:04,500
So we know the route.

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We know the solutions.

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The quadratic formula tells us
that the two solutions-- there

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are always two,
but they could be

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00:02:12,570 --> 00:02:17,110
equal-- have this-- do you
recognize this expression?

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You see the damping
coefficient coming in.

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You see this all
important square root,

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which tells us, depending
on whether B squared is

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bigger than 4AC, B
squared is equal 4AC,

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B squared is smaller.

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So smaller B means less damping.

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And it would be
called underdamping..

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So here are the possibilities.

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B could be 0.

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00:02:49,810 --> 00:02:53,760
That puts us back
in a previous video

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00:02:53,760 --> 00:03:01,110
when the solution was a pure
complex exponential, pure sine

51
00:03:01,110 --> 00:03:02,660
cosine.

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There was no damping.

53
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It just oscillated forever.

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That's the one we know.

55
00:03:09,850 --> 00:03:15,220
Now, the new ones with a B are
B squared could be smaller.

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So that's only a little damping.

57
00:03:18,520 --> 00:03:22,100
And in that case, what does
the solution look like?

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If B squared is
smaller than 4AC,

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I have to always look back here.

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So I have something negative.

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Something negative,
a minus B over 2A.

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And then plus or
minus-- and what's

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important point about this guy?

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B squared is smarter than 4AC.

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This is negative inside.

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So its square root is
an imaginary number.

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So the plus or minus,
there's an imaginary number.

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A little different from
the natural frequency.

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It's called the damp frequency.

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Not quite as frequent.

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The damping slows the
oscillation down a little bit,

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but it brings in
exponential decay.

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I'll draw a picture this time
or next time of the solution,

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e to the minus st.

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But you see, e to the
minus st is decaying to 0.

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It's like a spring that's
slowly winding down to 0.

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But it's oscillating
as it does it.

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OK.

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Now, we've got two more
possibilities here.

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And these are just
fun to get straight.

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There's the next one.

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When B squared is exactly 4AC.

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Exactly 4AC.

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B squared over 4AC, that's going
to be the critical ratio here.

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And here that ratio is 1.

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And in that case, that
square root is nothing.

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The square root of B
squared minus 4AC is 0.

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So we get s1 and s2 equal.

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Two equal frequencies.

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Real numbers.

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They're the minus B over 2A.

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So s1 and s2 are
both minus B over 2A.

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That's just the edge between
the decay with oscillation

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from underdamping.

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And the opposite
extreme is overdamping.

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Overdamping, when B
squared is bigger than 4AC,

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then our formula-- we're
always looking back

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at that quadratic formula.

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B squared bigger than
4AC, this is real.

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Square root of a
positive number.

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It's perfectly real.

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So in that case, we
have two, s1 and s2.

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Oh, let me draw
you some pictures.

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I'll draw you the
parabola, the As squared.

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So this is a graph
of As squared.

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Let me plot As squared
plus Bs plus C.

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So in this overdamping case,
we might have a lot of damping.

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B is pretty large.

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And that would be--
this direction is s.

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And I'm graphing here s squared
plus-- well, let me see.

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I think probably plus 2s.

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And now I'm going to
choose this-- oh, actually,

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plus 0 to get that one.

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So this has a root at--
this is overdamping.

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Overdamping.

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A big number there.

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C is 0.

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No mass at all.

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No stiffness at all.

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So this parabola has a root
at 0 and a root at minus 2.

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That would be an extreme case
with no stiffness whatever.

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Now, let me add some stiffness.

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So what happens if I
change the constant term?

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It lifts the graph.

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It lifts the graph.

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So let me make it 1.

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So I move the whole
graph up by 1.

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So you see that as I
move it up, these roots

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will come closer and closer.

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And at the right point,
when I move it up to 1,

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what's going on there?

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Critical damping.

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Critical damping.

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This is the case s
squared plus 2s plus 1.

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Which is exactly s plus 1 twice.

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And it has that repeated
root, which we always

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recognized as the marginal
case when it doesn't go below,

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but it doesn't
stay above either.

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It hits here twice.

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So the root is minus 1 twice.

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That's critical.

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And now what?

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Increase this constant
C even further.

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You'll lift the graph,
lift the parabola up here.

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Lift it by one more, let's say.

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And what's going on there?

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So this would be s
squared plus 2s plus 2.

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And you see what that
graph is telling us.

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It's telling us there are
no roots, no real roots.

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00:09:06,370 --> 00:09:11,250
This is a real graph in the
real plane, and it's never 0.

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This is s plus 1.

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This is the case when I have
s plus 1 squared plus 1.

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Can't be 0.

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Never 0.

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So this has-- sorry.

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With the 2 there,
it has damping.

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It has damping, but
it's underdamping.

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The roots are complex.

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So this s plus 1 squared is 0.

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00:09:55,590 --> 00:09:57,290
What are the roots of that?

161
00:09:57,290 --> 00:10:01,980
s is minus 1 plus or minus i.

162
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That's the case of underdamping.

163
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So let me write that.

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We see so much
from these graphs.

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So this was a graph
in which I changed

166
00:10:19,810 --> 00:10:25,760
the constant term, the
capital C, the stiffness, k.

167
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So that lifted the graph.

168
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And eventually the
roots were complex.

169
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They weren't real anymore,
because the parabola

170
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didn't cross the axis anymore.

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The roots are where
it crosses the axis.

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So that's one.

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I'll draw in the next
video another picture.

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Maybe change B.
Change the damping.

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00:10:52,850 --> 00:10:55,250
So what would happen if
I change the damping?

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If there's no damping, the
roots are pure imaginary.

177
00:11:01,801 --> 00:11:02,300
Yeah.

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00:11:02,300 --> 00:11:05,620
Let's just go back through
those four possibilities,

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because that's what
you have to learn.

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00:11:09,300 --> 00:11:13,500
If B is 0, there's
no damping at all.

181
00:11:13,500 --> 00:11:16,340
And we're back in
the pure oscillation,

182
00:11:16,340 --> 00:11:20,250
where is pure imaginary.

183
00:11:20,250 --> 00:11:23,490
We're just seeing
cosines and sines.

184
00:11:23,490 --> 00:11:29,990
Then if we add a little damping,
there appears a decay turn.

185
00:11:29,990 --> 00:11:38,860
If we add-- I could even
draw these possibilities.

186
00:11:38,860 --> 00:11:44,830
Let me draw the solution
y of t, against t.

187
00:11:44,830 --> 00:11:46,020
OK.

188
00:11:46,020 --> 00:11:50,760
So underdamping--
the solutions say

189
00:11:50,760 --> 00:11:57,591
it starts there-- will
oscillate, but decay.

190
00:11:57,591 --> 00:11:58,590
So this is underdamping.

191
00:12:03,350 --> 00:12:07,250
And you remember
when that happens?

192
00:12:07,250 --> 00:12:11,550
That's when B is
present but smaller.

193
00:12:11,550 --> 00:12:15,080
B squared is smaller than 4AC.

194
00:12:15,080 --> 00:12:17,960
Overdamping, if it
starts, let's say,

195
00:12:17,960 --> 00:12:23,140
here, it will going
to 0, probably

196
00:12:23,140 --> 00:12:25,210
with no oscillation at all.

197
00:12:25,210 --> 00:12:26,720
It could have one oscillation.

198
00:12:26,720 --> 00:12:28,450
Depends how it starts.

199
00:12:28,450 --> 00:12:30,440
So ooh, let me
make this picture.

200
00:12:30,440 --> 00:12:33,550
So that one is the overdamping.

201
00:12:33,550 --> 00:12:36,290
This guy is the underdamping.

202
00:12:36,290 --> 00:12:39,200
Underdamping still
has oscillations.

203
00:12:39,200 --> 00:12:42,730
Overdamping has, at most, once.

204
00:12:42,730 --> 00:12:44,340
It could cross once.

205
00:12:44,340 --> 00:12:51,900
But overdamping is B
squared bigger than 4AC.

206
00:12:51,900 --> 00:12:52,770
OK.

207
00:12:52,770 --> 00:12:53,950
There's more to do.

208
00:12:53,950 --> 00:12:59,870
This is so central that you have
to think about the ratio of B

209
00:12:59,870 --> 00:13:01,400
squared to 4AC.

210
00:13:01,400 --> 00:13:03,280
That has its own name.

211
00:13:03,280 --> 00:13:09,890
And it's a damping ratio, or
the square of a damping ratio.

212
00:13:09,890 --> 00:13:13,500
And we identify all
these solutions.

213
00:13:13,500 --> 00:13:18,040
Does everybody know where the
solution will be when B is 0?

214
00:13:18,040 --> 00:13:19,580
No damping at all.

215
00:13:19,580 --> 00:13:21,832
In that case, it
will just oscillate.

216
00:13:24,830 --> 00:13:29,950
So my picture is getting, you
could say, a little messed up.

217
00:13:29,950 --> 00:13:32,100
But this pure
oscillation would be

218
00:13:32,100 --> 00:13:34,676
the B equals 0 with undamped.

219
00:13:39,370 --> 00:13:39,870
OK.

220
00:13:39,870 --> 00:13:43,300
That's a first look at
quadratic equations.

221
00:13:43,300 --> 00:13:48,080
As squared plus Bs plus C. It
has the big name characteristic

222
00:13:48,080 --> 00:13:50,450
equation, but you
could see, it's

223
00:13:50,450 --> 00:13:55,110
the fundamental equation for
a second order differential

224
00:13:55,110 --> 00:13:56,250
equation.

225
00:13:56,250 --> 00:13:57,470
So we'll see more of it.

226
00:13:57,470 --> 00:13:59,020
Thanks.