GILBERT STRANG: OK.
This is the second
video for Chapter 3.
And it's going to
be pictures again.
But it's pictures for a
second order equation.
And I'll make them--
these will be nice.
We'll know formulas here.
These will be
constant coefficient,
linear second order equations.
And we know that
the solution-- there
are two special
solutions, e to the s1 t
and e to the s2t,
two null solutions,
and any combination
is a null solution.
So we're talking
about null equations,
0 on the right-hand side.
And we just want to
draw that picture that
goes with solutions like that.
So here is the magic word,
phase plane, phase plane.
We're going to draw the
pictures in a plane.
Because that's what
a blackboard is.
And the axes we'll choose
will be y and y prime, not t.
You'll see how t, time,
comes into the picture.
But we have the two axes
will be y and y prime.
So I had to figure
out what y prime was.
It just brings down
an s1 from that term,
and brings down an
s2 from that term.
And now here's the example.
Here is the first example.
So I took this
particular equation.
Notice that the damping
term is negative.
I have negative damping.
This will be unstable.
Solutions will go
out to infinity.
And I can find those solutions.
Because you know that
I look for e to the st.
I always look for e to the
st. I plug in e to the st.
I get an s squared
from two derivatives,
minus 3s from one derivative,
plus 2 equaling 0.
I factor that, and I find
the 2s1 is 1, and s2 is 2.
And now I'm ready for
the phase plane picture.
OK.
Phase plane picture, so
here are my solutions.
s is 1 or 2.
Then the derivative
has a 1 or a 2.
And here's my plane.
Here's my plane.
And I want to draw on
that the solutions.
These solutions, I
actually have formulas.
I just want to draw them.
So I'm plotting.
One example would be that
c1 be 1, and let c2 be 0.
So that's gone.
c1 is 1.
I just have that picture.
What kind of a picture
do I have in the phase
plane, in the y, y prime
plane, when that's y
and that's y prime?
Well, those are equal.
So y equals y prime
for that solution.
y equals y prime along
the 45-degree line.
It's just like y equal x.
y prime is y.
And what's happening
on this 45-degree line?
The solution is this solution,
is going straight out the line.
As t increases, y and
y prime both increase.
I go out.
This is t going to infinity.
And what about t going
to minus infinity?
Because we got the
whole picture here.
When t goes to minus infinity
that goes to 0, that goes to 0.
Here is the point where
the universe began.
The Big Bang is right there
at t equal minus infinity.
And as t increases,
this point, y, y prime,
is traveling along
that 45-degree line.
Because y equals y
prime, and out there.
And what about the
rest of the line?
Well, if c1 was negative,
if c1 was negative
I'd have a minus there,
and a minus there.
I would just have minuses.
And I'd be going out that line.
Well, that's one line in my
whole plane, but not all.
Now let me take as a
second line c1 equals 0.
So nothing from e to the t, and
let me take y as e to the 2t,
and y prime then
would be 2e to the 2t.
OK.
What's happening in the phase
plane for this solution,
now looking at this one?
Well in this solution, in this
case, y prime is 2 times y.
y prime is 2 times y.
So I'm staying on
the line y prime,
where y prime is 2 times y.
It's a steeper
line, steeper line.
So that was the case, this
was the line where c2 was 0.
There was no e to the 2t on
that first line that we drew.
In the second line
that we drew, c1 is 0.
There's no e to the t.
Everything is in e to the 2t.
So now c1 is 0 on this line.
OK.
And we just go out it.
As t increases, y
prime increases faster.
Because of the factor 2.
So it goes up steeply.
And it goes this way.
When c2 is negative, if I
took a minus and a minus,
I would just go down the
other way on the same line.
And this is still the Big
Bang, t equal minus infinity,
where everything starts.
OK.
So that is two lines,
the two special lines
in the phase plane.
But now I have to draw
all the other curves.
And they will be curves.
And where will they come from?
They will come
from a combination.
So now I'm ready for that one.
Let me take the case
c1 equal 1, c2 equal 1.
Yeah, why not? c1 equal 1.
So I can erase c1.
c2 equal 1, I can erase c2.
And now I have another
solution, y and y prime.
And I want to put it
in the phase plane.
So at every value of t, at
every value of t that's a point.
That's a value of y.
This is a value of y prime.
I plot the points y and y prime,
and I look at the picture.
And again as t
changes, as t changes
I'll travel along the solution
curve in the phase plane.
I'll travel along.
As t changes, y will
grow, y prime will grow.
I'll head out here.
But I won't be on that straight
line or that straight line.
Because those were the
cases when I had only one
of the two solutions.
These were the
special solutions.
And now I have a combination.
So what happens as
t goes to infinity?
As t goes to
infinity, this wins.
As t goes to infinity,
the e to the 2t
is bigger than e to the t.
So this is the larger term.
So it approaches.
This curve now will
approach closer and closer
to the one when the
line with slope 2.
The 2 will be the
winner out here.
But at t equal minus
infinity, near the Big Bang,
at t equal minus infinity, e
to the 2t is even more small.
So at t equal minus infinity,
or t equal minus 10,
let's say, this is
e to the minus 10.
This would be e to the minus
20; very, very, very small.
These would win.
So what happens
for this solution
is it starts out along the line
given by the not-so-small t,
the not-so-small exponent.
It starts up that line.
But t is increasing.
When t passes some point,
this 2t will be bigger than t.
And it will, I guess,
at t equals 0, 2t
will be bigger than t.
And from that point on, from
the t equal 0 point-- oh,
I could even plot
the t equals 0.
So at t equals 0, y
is 2 and y prime is 3.
So at 1, 2, 1, 2, 3;
somewhere in there.
So you see, the curve starts up
along the line where e to the t
is bigger.
They have the same size at
t equals 0, both equal 1.
This is at t equals 0.
And then for large
times, this one wins.
So I approach that line.
I don't know if you
can see that curve.
And I don't swear to the
slopes of that curve.
But in between in
there is filled
with curves that start
out with this slope,
and end with that slope.
And the same here, it'll
start with this slope.
But then go-- probably
this is a better picture.
Yeah.
That's a better picture.
Yeah.
It will just go up with slope.
At the end it will have
slope 2 going upwards.
Yeah.
That looks good.
Well, you could say I only
drew part of the phase plane.
And you're completely right.
If I start somewhere here, what
would you think would happen?
What would you
think would happen
if I start with that value of y
that much, and that value of y
prime?
It would have some mixture of--
there would be a c1 and a c2.
So the other curves
that I haven't drawn yet
come from the other c1 and c2.
I've done c1 equal
1, and c2 equal 1.
And c1 equal c2 equal 1.
But now I have many
more possibilities.
And what they do is they will--
so suppose I start there.
It will approach--
this is the winner.
This is the winner.
Where c1 is 1, where
this is happening,
there is the winner
for large time.
So all curves swing up
toward parallel to that line.
Or down here, they swing
down parallel to that line.
So things here will
swing down this way.
That's the phase plane.
May I do one more example to
show that this was a source?
This is called a source.
Because the solution
goes to infinity.
Wherever you start, the
solution goes to infinity.
It's unstable, totally unstable.
Now if I change to
a positive damping,
then I would have
a plus sign there.
These would be plus signs.
I would have s equal minus
1, or s equal minus 2.
So with positive damping,
I damp out naturally.
And this picture
would be the same,
except all the lines
are coming in to 0, 0.
The solutions are damping to
0, 0; to nothing happening.
So I just track the same lines,
but in the opposite direction.
So instead of this
being the Big Bang,
it's the end of the
universe, t equal infinity.
OK.
I'm up for one more picture
of this possibility.
And let me take the equation
y double prime equal 4y.
So my equation will be s squared
equal 4, s equals 2 or minus 2.
And when I draw the phase
plane and the solutions,
the solutions will be c1 e to
the 2t, and c2 e to the minus
2t, from a 2 and a minus 2.
That's the solution we all know.
And now I should
compute its slope.
y prime will be 2c1 e to the
2t minus 2c2 e to the 2t.
And now you just want me
to draw those pictures.
You just want me to
draw those pictures,
and let me try to say
what happens here.
This is a saddle point.
It's called a saddle, when we
have in one direction things
are growing, but in the
other, things are decreasing.
So most solutions,
if c1 is not 0,
then the growth is going to
win, and that will disappear.
But there is the
possibility that c1 is 0.
So there will be one
line coming from there.
There will be one line
coming from there.
Maybe I can try to draw that.
Again, I'll draw that
pure line, where c1 is 0.
So that pure line is coming.
These are minuses here.
So that line is coming
in to the center.
So that's why we have a saddle.
We approach a saddle if along
this where this is minus 2
of that, so I
think it would be--
so it's a slope of minus 2.
So I think a slope like
that, so again this is y.
This is y prime.
This is the slope of minus 2.
And that's this curve.
So it will be very exceptional
that we're right on that line.
All other points, all other
curves in this phase plane,
are going to have a
little c1 in them.
And then this will take over.
And that gives us, as we
saw before, this slope.
This is 2 times that.
So that line is where
everybody wants to go.
And only if you start
exactly on this line
do you get this picture, and
you come into the saddle.
Instead of the Big Bang,
or the end of the universe,
this is now the
saddle point, where
we come in on this most
special of all lines,
coming from this picture.
But almost always this
is the dominant thing.
And we go out.
So if I take a typical
starting point,
I'll go out this like
that, or like this, oh no.
Yeah, no.
I'll go out.
It'll have to go out.
So if I start anywhere
here, these are probably
they're hyperbolas going
out in that direction.
I don't swear that
they're hyperbolas.
Here again we might start in.
Because we have
big numbers here.
But then e to the t takes over.
And we go out.
So those go out.
These go out.
And these go out.
So this is the big line.
That's the line
coming from here.
And that's where
everything wants to go,
and everything eventually
goes that way, except the one
line where c1 is 0.
So this dominant term
is not even here then.
And then we should
become inwards.
So that saddle point
is the special point
where you could go out,
if you go the right way.
Or you could come in, if you go
the other special, special way.
OK.
So that sources,
sinks and saddles.
And I still have to
draw the pictures, which
involves spirals that
come from complex s,
where we have oscillation.
That'll be the next video.
