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PROFESSOR: The isoclines applet.
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Let's explore
graphs of solutions
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of differential equations using
direction fields and isoclines.
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The first thing to do
is to choose an equation
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from the pull-down menu here.
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And for this
demonstration, I will
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choose the
differential equation y
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prime equals y squared minus x.
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In the window at left, you
can see the direction or slope
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field drawn.
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This is a representation of
the differential equation.
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When I move the cursor
over this window,
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a readout of the x-
and y-coordinates
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shows up on the right-hand
side of the screen.
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This is very useful in
making measurements.
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If I click on the
graphing window,
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the solution through the point
that I've clicked on is drawn.
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This is an interesting
function, probably one
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you've never seen before.
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Certainly, it's not an
elementary function.
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Clicking some more points
makes more solutions appear.
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In fact, every
point on this plane
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has exactly one
solution through it.
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This is the meaning of the
existence and uniqueness
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theorem for
differential equations.
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Now, let's clear the solutions
using this button down here,
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Clear Solutions button.
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You see a blank screen again.
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You can control how the slope
field looks by using this Slope
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Field toggle here.
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You can make them
blank, or bright
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for a display, for example.
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Or the way it was originally.
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And I think I'm going to
turn it off altogether,
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so now we're faced
with a blank screen.
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Now, how would you go about
drawing some solutions
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to this differential
equation by hand just knowing
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the differential equation
and this blank screen?
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Well, isoclines give you
a good way of doing this,
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and they reveal things about
the qualitative behavior
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of the differential
equation as well.
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An isocline is the subset
of the plane where the slope
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field takes on a given value.
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I can choose that value,
m, using this slider here.
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And when I click
here and move this,
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the isocline with
the given value of m
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is drawn in yellow
on the screen.
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And you can see the value of
the slope field drawn as well.
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So here is the value is
1, and the slope field
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is given by little intervals
of value 1, of slope 1.
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I can choose that value
using the slider marked m.
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If I click on the
handle and drag it,
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you see the isocline for
the corresponding value
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of m drawn on the screen.
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And on the isocline is also
drawn the direction fields.
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So when m is 2, the
direction field has slope 2.
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And when I drag the
slider back down to 0,
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the isocline value 0 has a
horizontal direction field
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marked along it.
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Each one of these
is a curve where
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y squared minus x equals m,
or x equals y squared minus m.
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This is a parabola
lying on its side
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with the vertex at
x equals negative m.
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Let's draw the isocline
for value 1, for example.
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I've now released the
mouse key, and the isocline
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is left behind.
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And I can easily draw
in some other isoclines
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as well by clicking on
different values of m
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and releasing the mouse key.
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So once you've drawn several of
these isoclines on the plane,
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it's pretty easy to envision
what the solutions will
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look like to this curve.
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You just have to thread your way
along the part of the direction
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field that you've drawn.
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I can check that by
clicking on the screen
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and drawing a solution in.
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This is quite easy
to do by hand.
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It's easy to draw the
isoclines and then
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sketch a solution accordingly.
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But you can see other things
as well from isoclines.
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I'm going to clear all of
these and redraw the m equals
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0 isocline on the screen.
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And I think I'm going to
redraw the slope field as well.
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Now, critical
points of a function
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occur where the
derivative is equal to 0.
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And the derivative is exactly
what we know about the solution
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to a differential equation.
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So the critical points,
the minima or maxima
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of solutions to this
differential equation,
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occur when the solution crosses
the m equals 0 isocline, also
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known as the nullcline.
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All maxima and minima of
solutions to this differential
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equation occur along this
particular yellow parabola.
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I can check that by
drawing in some solutions,
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and you can see that these
functions have maxima which
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occur just along that parabola.
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If the solutions
miss the parabola,
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they don't have any
critical points.
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Another thing you can
see from this picture
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is that apparently
many of the solutions
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to this differential
equation cluster
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near to this branch
of the nullcline.
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And maybe we can see why this
is using the isocline picture.
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I'm going to clear the solutions
now to make the picture clearer
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and draw in one more
isocline, namely
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the m equals minus 1 isocline.
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Now, suppose that a solution
finds itself below the m
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equals minus 1 isocline.
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So it's in here somewhere.
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Can it ever cross the m
equals minus 1 isocline?
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Once it's below it,
can it ever cross it?
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Well, it's below it.
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So if it crosses the m
equals minus 1 isocline,
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it must cross it
with a slope which
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is bigger than the slope
of that yellow curve.
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But when it crosses
it, it also has
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to cross it with slope
minus 1, because this is
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the m equals minus 1 isocline.
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But the slope of this yellow
parabola is bigger than minus 1
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along here, and so the solution
curve can never cross it.
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Similarly, suppose that
you have a solution which
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is above the nullcline.
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Can it ever cross the nullcline?
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Well, if it crosses
the nullcline,
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it must cross it from above,
so when it crosses it,
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its slope must be less than the
slope of the nullcline, which
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is negative as you can see.
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But when the solution
crosses the nullcline,
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it must cross it with slope 0.
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And so, that can't happen.
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And so you see, if a solution
is between those two isoclines,
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then it stays between
them forever more.
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It's trapped between them.
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This is called a funnel.
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It's trapped between
these two things,
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and gets closer and closer,
because these two isoclines
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become asymptotic
as x gets large.
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These are called fences as well.
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Once a solution is in
here, it can't cross
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either of these two fences.
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This let's us estimate the
value of solutions for large x.
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These solutions,
anyway, for large x.
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For example, if x
is equal to 100,
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the solution has to be
bigger than the value
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of this parabola, which is minus
10, and less than the value
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along this parabola, which is
minus the square root of 99.
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So you get a very good
estimate for the value
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of solutions for large x from
these kinds of considerations.
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One more thing you can see
from this applet is this.
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If a solution is well
above the nullcline,
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it gets caught in
this powerful updraft
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and goes off to infinity.
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In fact, all these
solutions become tangent
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to vertical lines.
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They don't continue
for all large x.
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They blow up in finite time.
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On the other hand, if you're
just a little bit smaller
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than these solutions,
then you get
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solutions which
cross the nullcline,
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get trapped into
this parabolic region
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and fall down in between
our funnel, down here,
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and so become asymptotic to
minus the square root of x
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when x gets large.
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There's just one
solution which doesn't
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do either of these very
different behaviors,
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and it's right along here.
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It becomes asymptotic
to the positive branch
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of the parabola, and
every solution above it
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blows up in finite time,
and every solution below it
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falls down and becomes
asymptotic to minus
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the square root of x.
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This special solution-- which
doesn't do either of those two
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behaviors, but
continues to exist
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for all positive
values of x and become
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asymptotic to the
square root of x-- this
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is called the separatrix.
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It separates solutions showing
two very different types
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of behavior, the ones that fall
down and the ones that blow up.
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Well, these are just
a few of the things
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you can understand
using this applet.
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Play with different menu items.
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Open a copy of this applet
in your browser window.
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Play with some other menu items.
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Maybe the default item
at the bottom of this
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pull down menu down here.
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Are there funnels?
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Are there separatrices?
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What happens to solutions as x
goes to minus infinity rather
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than x equals plus infinity?
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Where are the critical points
of solutions of this equation?