This content introduces slope fields as a visual method to approximate solutions for differential equations, especially when exact analytical solutions are difficult or impossible to find. It emphasizes that slope fields represent the general solution by showing the slope of solution curves at every point in the plane, allowing for the identification of particular solutions given an initial condition.
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Hey, we're almost done exploring what
differential equations are and what the
solutions look like. We're almost done
with that and we're going to get moving
on in the next few videos here like
after three or four videos from now on
the process of solving differential
equations like what techniques we have
and then we're going to explore how to
use those in applications and then some
ideas of
some different ways to solve them. But
for right now, uh, one idea that I want
to share with you is is this idea of
slope fields. What a slope field is, why
we might have slope fields, and this
kind of breaking the pattern in our
heads that we have to have um an exact
solution all the time in in real life
because that's not true. So, a lot of
differential equations use
approximations because there aren't any
good techniques to solve some of these
things we're going to come up against.
That's one of the ideas of slope fields.
So in this video, what I want to get at
is what a slope field is, how a slope
field can help us identify a solution
even if we can't explicitly define it.
And that's important. So here's a slope
field. Uh they'll go through a problem,
a solution, an idea, and a process of
how we're going to go about it. So
here's the problem. Lots of times
differential equations can't be solved
if they have this dependent variable y
in them. You look at it, you go,
there's nothing we have. It's not
separable. We'll talk about separable.
It's not linear. Talk about linear. It's
not exact. It's not a substitution we
can make. It's not homogeneous. We go.
What? Well, I don't understand those
things. You're not supposed to right
now. But we're going to get to those
techniques of solving differential
equations as we move forward. So, don't
worry about it. But the point right now
is that sometimes we come across these
differential equations that have y in
them that we're not going to be able to
do anything with. and you look at it,
you go, I there's nothing I can do. So,
what what's a possible solution to that?
That's what a slope field and and some
other techniques help us with. So,
here's an example of the derivative of y
with respect to x. So, a differential
equation has a first derivative, a first
order differential equation has x^2 + 2
y^2 and we go, that's got a y in it.
There's nothing that we know of that's
going to help us to solve that. I
there's nothing to do. So sometimes you
run into that in differential equations
where there's nothing that we can do to
get an exact answer, an exact general
solution or even a particular solution
that's going to fit that. So one
solution to our problem is we can often
times approximate solutions and that
works with for some practical purposes
and a lot of people in math when they
get here they don't like that. I didn't
like it. I still don't really like it
that there's not a solution. You know,
when we have linear equations, you solve
it and you get one answer or you get no
solution or infinite solutions, but we
we get an answer and there's a process
to doing that. Well, with the difference
of equations, sometimes there's not. And
so, we have to get kind of used to this
idea that not all the time do we get the
answer that we're looking for by a
process that's outlined like step one,
step two, step three. It it doesn't
exist in some cases. So, we have these
techniques to approximate and it works
in real life really well. So, we say,
well, this this Huawei can't find an
actual solution. We can approximate it.
It's going to work just fine for us. And
if you remember, we we talked about that
a few videos ago where we said that um
that these approximations are good
enough that that these approximations
are something that that we can model
life on and when we get too advanced in
our equations, we can't solve them
anyway. So
we have this trade-off. If we start
limiting variables, well then we are not
perfectly representing our situation.
But if we perfectly represent our
situation, then we we can't solve it. So
sometimes in making an equation better,
we limit our ability to solve it. And
that's where approximations can come in
and help us and say, well, I can't
particularly solve this, but it's good
enough for the the real life application
I'm trying to represent. So here's our
idea. So, so again,
sometimes you can't solve it with the
techniques I'm going to teach you.
Here's a solution. We have
approximations. Here's the idea on how
to get that approximation. One idea,
it's called a slope field. So, here's
what this idea does.
If we know that first derivatives
represent slopes, we know that a
derivative is a rate of change. A first
derivative represents a slope of a
function that we're dealing with. So, a
So if we have a first derivative, a
derivative of y with respect to x,
derivative of the dependent variable
with respect to the independent
variable. If we have that, solve for
some function that has both x's and y's
in it. What that means is we have a
formula to find the slope. That's kind
of cool. This represents the slope. So
if this is the first derivative, this is
the slope. That's our formula. So our
first derivative, our slope is
represented by this formula with x's and
y's. That's neat. So this means that we
More specifically, this represents slope.
So the first derivative represents
slope. So we have a formula. Find the
slope at any point at any point that we
Why? Why? Why? Why does that work? Well,
if this is our slope and it's based on
x's and y's points have an x and a y.
by just plugging it into the function.
And that should be pretty
straightforward to us. If a first
derivative means slope and this slope
equals some function in x and y given
any point boom x and y we can plug that
point into this function and find a
slope at that point provided that it's
continuous at that that point. So
provided that we we're not at an end
point provided that we don't have a gap
or a hole or it's not defined. So that
that's important. We'll get into
uniqueness and existence a little bit
So, short recap. We're almost done. I'm
going to get to two examples to show you
what slope fields are. They take a while
to create. They're not hard. A lot of
people use computer graphing to do this
or a calculator. I'm not going to be
showing you that. Um, of course, my
ideas here are are to get you to
understand the concept, not just to be
able to do it, but to understand what's
going on. And I I like to do that by
hand when I go through this. So, slope
fields problem. Uh, not we can't solve
them all. Solution maybe we can
approximate them. My idea is hey if
derivatives represent slopes and you
have a differential equation with a
derivative in it and we can solve it for
a function of x and y then we have a
formula to find slope at any point. This
is slope. That's the formula. Just plug
in the point to our function and you're
going to find the slope at whatever
point xy you just plugged in. So here's
our process. What we're going to do is
is we're going to look at this. We're
going to make a whole bunch of little
lines that represent slopes at points on
It's not as hard as you might think cuz
you're going to have this tons of
slopes, but there's a pattern to it. I'm
going to show you. So, make a graph. So,
xy plane that has a lot of lines that
represent little slopes. So, they look
like unit vectors almost without the
arrow. of if you've had calc 3 and you
have you've had vectors. So you see
those little unit vectors all over the
place. They really look like that
without the arrows at points on xy. They
just give a direction that your line is
going. Well, that's what slope does,
right? It gives the rate of change at
any point on a on a function. And so our
idea is if we make this graph with a
whole bunch of little lines that say,
hey, at this point the function should
be going like this. Well, then we can
put all of that together and given a
Then we can find an approximate curve
whose slopes
fit that
whose slopes fit. And what we mean by
that is it is this. So here's the
general idea and it's going to seem
vague because we haven't had any
examples, but here's the idea. You're
going to take this function, this
differential equation that represents
slope. Slope is a first derivative. And
you're gonna take every point on the xy
plane. And that sounds hard. Uh take
every point and plug it in and say,
"Well, at this point I get this slope.
At this point I get this slope. At this
point I get this slope." And you're
going to make this huge xy plane.
There's going to be slopes everywhere.
These little lines that represent
direction at every point. The slope can
be given at any point just by plugging
it in. So with the whole xy plane, we
could potentially plug in every point.
Can't blame every point but like the
point 3 comma 4 the point 1 comma 2 0 0
we can find the slope for any of those
points where we have a defined function.
So where it's continuous where it's
differentiable we can find the slope of.
So we're going to do that. We're going
to take the whole xy plane. We're take
every point on it that we can find and
we're going to find the slope at those
points. It's going to look crazy. We
have all these little directions all
these little slopes on this on this this
plane. Then if I give a specific point,
well, we can say there's this is the way
the curve has to go or has to travel to
make it through that point and to fit
all of these slopes. So given a point or
an initial condition, that's what we're
talking about. Then we can start
approximating a curve. So here's what
you need to know about these slope
fields that we're going to create. When
we create a slope field, so all these
slopes on the plane, it's a general
solution. It says that this is
everything that could happen. These are
how all the curves would have to look to
go through these these slopes to make
sure that we're fitting the slope at
every single point. If you want one
specific curve, a particular solution,
like we studied before, you need an
initial condition. You need a point to
go through. So, long story made short,
sometimes stuff don't work. We need
approximations. One way is to find the slope
slope
at every point. Put it on a graph and
then that represents your general
solution. That represents all the curves
that would have such slopes that would
have such slopes. Then what we do is we
say if we want a specific one, you just
need a point and that will limit the
curve. So we'll find a particular
solution that fits on our slope field
that also goes through that point and at
every point along that curve, we're
fitting the slopes. That's the idea.
Well, what I want to do now is show you
two examples on how to create them. And
then all of this is going to make a lot
more sense because as we go through,
I'll be explaining, hey, remember how
this makes slope and remember how we
need a point and now we can go ahead and
see a specific curve. So, I'm going to
do that. Uh we'll we'll do just two
examples and I'll walk you through how
to do that.
So, let's get started on our example.
We're going to build a slope field from
scratch. So, no computers, no
calculators because I want you to
understand what's going on here. they'll
do it for you. And I'm sure that you're
going to use it in whatever class you're
in. You can get a computer and that's
the a great way to build slope fields
when you have some difficult
differential equations. So here's our
problem. Again, sometimes these things
aren't solvable. I'm not saying this
one's not. I'm saying I'm giving you an
easy example here so we can see what a
slope field does. So whether there's a
te technique or not to solve that
explicitly, I don't care right now. What
I want is your idea of slope fields to
be built in your head so you so you
understand how they work. So here's our
point. If that right there means a
slope, then that's the formula for the
slope. So at every single point that we
have x, comma y, we can find the slope
at that point of our general solution.
our slope first derivative is just x +
y. That means that if we have our y
we can make a table for slope really,
really easily. All we do is we have to
add up our x plus our y value. It says,
hey, take the x value plus y value. It's
going to give you the slope. We're going
to make a table up like that. It looks
like it's going to take a long time, but
there's a lot of good patterns here. So,
we're going to I'm going to show you how
to find the patterns. I'm going to show
you how to write it really quickly. Then
we're going to take our slopes over, put
them on our xy plane, and that will
create a slope field. We'll talk about
it in just a minute. Now, you can do
this however you want. I prefer to put
my y values up top and my x values on
the side. You can do it differently if
you'd like to, but the point is try to
find a pattern and then and then utilize
that pattern. So, if you want to start
here, you can. I might start looking for
something like um well, if I'm if I'm
adding up some numbers here, I know that
-4 + 4 is going to give me zero. Are you
seeing that? Four for my x + -4 for my y
is going to give me zero. But so is 3 +
-3 and 2 + -2 and 1 + 1 and 0 plus 0 and
negative 1 plus 1 and so all this
diagonal is zeros. The x value is four,
the y value is4. When you add them to
find the slope which goes in this this field,
field,
zeros all the way along that diagonal.
You can check if you really want to, but
-4 + 4 is going to zero. We did that the
whole way. Now, your pattern is going to
continue diagonally for this whole rest
of it. So, if you find your first
diagonal, all you need to do is find one
number and all these are going to be the
Maybe try it for a couple of them. So
for these simple ones, we get these nice
diagonals. Do they all work like that?
No. But a lot of them do. And so we're
Also, you probably could make patterns
this way. So, 0gative 1, that's probably
-2, that's probably -3, that's probably
-4, that's probably5. Works that way, too.
And sure enough, -4 +4 is8. And we're
going to continue our patterns. I'll use
and we've already got half of our slope
field done. That's pretty nice. So, if
you're going to have to do this by hand,
most of the time you can find some nice
patterns. If you can't use a computer
you this is just to give you an idea of
what a slope field. I'm not going to
explore that with you. Um you can use a
computer or calculator to plug that in.
Uh usually every single textbook that
that's out there has some sort of a
graphing utility. You can walk yourself
through. They're not bad. Check them out
for yourself. Um but this this is just
to get the idea of what a slope field
does in your head. Same thing works
here. So if this is - 1 0, that's
probably one. Let's just check real
So now what we've done, we've taken this
thing, this differential equation, which
has a first derivative in it, which
means slope, and we have a formula to
find it. We just plug in every single
point for this finite graph that we
have. So it's it's limited, but you
could extend it, I suppose. And we find
the slope by just adding the x and the y
coordinates. That's for this specific
differential equation. Now, we take all
of these slopes and for point by point,
we're going to put just a little line on
this graph to represent the slope at
this point. It looks like a unit vector
without the arrows. It's giving you the
direction. That's what slope does. Gives
you the instantaneous rate of change or
the direction that the curve is going at
that point. What that's going to do is
fill out this entire xy plane with all
these sort of slopes that represent your
general solution. all the curves that
would fit these slopes. So, we'll get to
that right now. Okay. So, now that we
have this filled out with all of our
slopes at every point that we really
need on the xy plane, what we're going
to do is put those little lines at every
one of these points. I tried to do this
the first time without the the grid.
It's really hard. It's going to get all
disorganized. So, make yourself a grid
or get some some grid paper, some
graphing paper. That way you you know at
every single one of those intersections
of your x and your y that's where you're
going to put your little line. So when
we do this the this slope field what's
happening is that at every single
crosssection of an x and a y we now have
a corresponding slope. So we just have
to go along. I'm going to use my
diagonals. I'm going to start at 44 and
I'm going to make a diagonal. So 44 x =
4 and y =4 x = 4 y=4 I have a slope of 0.
So
if there's a particular solution that
hits that point, it is going to have a
slope of zero at that point. It's going
to be flat. So it's probably going to be
like a a relative max or a relative min.
It's going to bounce somehow. It's going
to do that. Now, because you have these
diagonals, just follow your pattern. At
three on the x and -3 on the y, three on
the x andg -3 on the y, we're also going
And we're going to have that at every
one of these intersections. Hopefully,
you can verify by looking here that
yeah, you know what? at -1 on the x and
one on the y.
1 x one on the y have a slope of zero.
And at -2
2 -3 3 and -4 4 those all have a slope
of zero. That's how a slope field works.
And we're going to take some time and
just fill this out. Maybe you can do it
as I'm doing it if you've drawn this. If
not, just follow me along. That's okay.
I have another one in just a minute that
you can do on your own if you'd like. So
be careful. It's kind of easy to get
these confused, especially confusing
your x with your y. So, I pause for for
just a minute every time I start a new
diagonal, and I make absolutely sure
that I have the right starting slope for
whatever point I'm on. So, I'm going to
go and do four on the x,
-3 on the y, and I'm going to put a
positive slope of one. So, four on the x,3
x,3
on the y. Four on the x, -3 on the y.
Now I'm going to see how that pattern
works. So the next one, three on the x,
-2 on the y. Three on the x, -2 on the y.
y.
And I'm going to try to give that
the same slope
And drawing that grid makes it just
infinitely easier. Not infinitely, but a
lot easier. Okay, let's keep on going.
Also, we can we can use this pattern
down here. So, as we're going on the x=
4 vertical line, so going up the y's,
our slope goes from 0 to 1 to 2 to 3 to
4 to 5 to 6 to 7 to 8. Here's what that
means. This was a slope of zero. 1 2 3 4
5 6 7 8. It's going to be really steep.
So as we climb along our y for a given x
value, our slope is increasing. So you
There's about a slope of two. So I'm
going to continue that diagonal. I'd
have a slope of 3 4 5 6 7 and 8. It
you're approximating here. Um so your
slopes are going to get really really
So zero, one, I followed it. Two, I
followed it. I know this is going to be three.
0 1 2 3 Looks pretty close. Here's going
to be four. So at four on the x and zero
on the y, I better have a slope of four.
Four on the x, zero on the y, I've got a
Next, I should have a slope of five. I'm
going up on that x value and my slope is increasing.
