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## AP®︎/College Calculus AB

### Unit 7: Lesson 4

Reasoning using slope fields# Approximating solution curves in slope fields

AP.CALC:

FUN‑7 (EU)

, FUN‑7.C (LO)

, FUN‑7.C.2 (EK)

, FUN‑7.C.3 (EK)

Given the slope field of a differential equation, we can sketch various solutions to the equation.

## Want to join the conversation?

- does slope field give the exact solution of differential equation?

if no then why do we draw slope field?(2 votes)- You are essentially correct. The slope field is utilized when you want to see the tendencies of solutions to a DE, given that the solutions pass through a certain localized area or set of points. The beauty of slope field diagrams is that they can be drawn without actually solving the DE. This is a very useful tool often employed by people who study mathematical biology, or changes in populations over time (due to predation, climate, etc). Slope fields allow these people to view the probable trends of a certain population based on its conditioning factors without actually solving their DE's.(25 votes)

- is it possible to find solution if point is not given ?(2 votes)
- Yes, in differential equations you can find what is called a "general solution" even if no initial conditions are given. If you do have initial condition then you can transform your general solution into a "particular solution".

With the slope field visualisation, you saw Sal draw several particular solutions, each one dependent of the initial conditions that he choose.(12 votes)

- How can a value be a solution for a differential equation. Didn't Mr.Khan mentioned the solutions for differential equations are functions or class or functions?(4 votes)
- Absolutely correct but it could also be a function that is not dependent on x. For example, the differential of y=3x+2 is simply y'=3, and so the value 3 is a solution for the differential equation(6 votes)

- sal took x=1 and y=6 so slope=-2 ? i am curios what does the slope=1.5342 or slope=pi,or golden ratio look ? sorry for my bad english , hope you got the point(2 votes)
- As the differential equation dy/dx is a function of y, plugging in the y-value 6 gives

dy/dx = 6/6 * (4-6) = 1 *-2 = -2,

the slope you mentioned. If you look at the point (1, 6) on the slope field diagram, you can see a short downward sloping line, of approximately slope -2.

If the slope were pi at a point, you would see an upward sloping line of approximately 3.14159... . We can solve for which points this would be at:

dy/dx = pi = y/6 * (4-y)

pi*6 = 4y-y^2 //multiply both sides by 6, then distribute on the right

0 = -1*y^2 +4* y - 6*pi //add 6*pi to the right side

If you solve the quadratic, you should get the y-values where the slope is pi. You can do this similarly for any slope value you wish.

I hope this helps!(8 votes)

- Can we get a solution in which for one x we get two or more y values . I mean not exactly function we normally see but a multivalued function.(4 votes)
- You may want to consult http://en.wikipedia.org/wiki/Differential_inclusion .

i think it is much-much forward compared to the above material. It nevertheless exists, so if you want it, go for it! :)(2 votes)

- well, actyally the dy/dx is the slope of the tiny straights we design? for example for x=1 y=1 dy/dx=1/2 we design a straight with angle 1/2 degrees??(1 vote)
- Not the angle, but the slope. The slope of the segment you draw at (1,1) would be 1/2. The angle would be arctan(1/2) from the horizontal.(8 votes)

- Wouldn't y=4 just be the critical point? Why would it be the solution?(3 votes)
- In this slope field, if we have Initial value y(0) = 2, similar to the Orange Dot, is our solution for the initial value y(0) = 2 ONLY the orange line? Not the asymptotes or any other curve? Is our solution for initial value y(0) = 5 ONLY the green line on top? The video should clarify this better.(2 votes)
- I'm pretty sure that it is a solution only for those initial values, as you seem to suggest.(1 vote)

- What would be a better way to create slope filed for our solutions or specifically how can i make and use slope filed tool in my mathematics study?(2 votes)
- 1. Cant we rather rearrange the terms and integrate?

2. Why do we need slope field at all? Now obviously, there are going to be infinite solutions, so, what is the point?(1 vote)- 1. You are right, we will integrate. definitely. And indefinitely too. :)

2. We may have cases where "only slope field exists", e.g. turbulence in any fluids. Your favorite math software will actually use Runge-Kutta approximation (RK4), that uses slope field data to derive your solution graphically. It is good to know what your software actually does before you use it.(3 votes)

## Video transcript

- [Voiceover] So we have
the differential equation, the derivative of y with respect to x is equal to y over six times four minus y. And what we have plotted right over here is the slope field or a slope field for this differential
equation and we can verify that this indeed is a slope field for this differential equation, let's draw a little table here, so let's just verify a few points, so let's say x, y, and dy/dx. So let's say we start with, I don't know, let's start with this point
right over here, one comma one, when x is one, and y is
one, well when I look at the differential equation,
1/6 times four minus one, so it's 1/6 times three, which is 3/6, which is 1/2, and we see
indeed on the slope field, they depicted the slope
there if a solution goes to that point, right at that
point, its slope would be 1/2. And as you see, it's actually
only dependent on the y-value, it doesn't matter what x
is as long as y is one, dy/dx is going to be 1/2,
and you see that's why when x is 1 1/2, and y is one,
you still have a slope of 1/2 and as long as y is one,
all of these sampled points right over here, all have a slope of 1/2, so that, just looking at
that, that makes us feel that this slope field is consistent with this differential
equation, but let's try a few other points just to feel a
little bit better about it, and then we will use a slope field to actually visualize some solutions. So let's say, let's do
an interesting point, let's say we have this point, actually no, that's at a half point, let's say we have this, let's see, let's say we
do this point right over here, so that's x is equal to
one, and y is equal to six, and we see the way the
differential equation is defined, it doesn't
matter what our x is, it's really dependent
on the y that's going to drive the slope, but
we have 6/6, which is one, times four minus six,
which is negative two. So it's negative two, so we should have a slope of negative two and it looks like that's what they
depicted, so as long as y is six, we should have
a slope of negative two. Have a slope of negative two, and you see that in the slope field. So hopefully you feel
pretty good that this is the slope field for
this differential equation, if you don't, I encourage
you to keep verifying these points here, but
now let's actually use the slope field, let's actually use this, to visualize solutions to
this differential equation based on points that the
solution might go through. So let's say that we have a solution that goes through this
point right over here. So what is that solution
likely to look like? And once again this is going
to be a rough approximation, well right at that point it's going to have a slope just as
the slope field shows, and as our y increases,
it looks like our slope, it looks like our slope... so at this point I should be, actually let me undo that, so if I
keep going up at this point when y is equal to two
I should be parallel to all of these segments
on the slope field that y is equal to two,
and then it looks like the slope starts to
decrease as we approach y is equal to four, and
so if I had a solution that went through this point, my guess is that would look something, and then now the slope decreases again as we approach y is equal to zero. And of course we see that,
because if when y equals zero this whole thing is zero so our derivative is going to be zero. So, a reasonable solution might look something like this,
so this gives us a clue of, look, if a solution
goes through this point, this right over here might
be what it looks like. But what if it goes through, I don't know, what if it goes through
this point right over here? Well then, it might look like... It might look like this
by the same exact logic. So it might look like
this, so just like that, we're trying to get a
sense, we don't know the actual solution for this
differential equation, but we're starting to get a sense of what type of functions or
the class of functions, that might satisfy the
differential equation. But, what's interesting
about this slope field, is it looks like there's
some interesting stuff if our solution includes points between where the y value is
between zero and four, it looks like we're going
to have solutions like this, but what if we had y-values
that were larger than that or that were less than that,
or exactly zero or four? So for example, what if
we had a solution that went through this point right over here? Well, that point right
over here, the slope field tells us that our slope is zero. So our y-value is not going to change, and as long as our y-value doesn't change, our y-value is going to stay at four, so our slope is going to stay at zero, so we actually already
found this is actually a solution to the differential
equation, y is equal to four is a solution to
this differential equation. So, y is equal to four,
and you can verify that that is a solution,
when y is equal to four, this right-hand side is going to be zero, and the derivative is zero
for y is equal to four. So that is a solution to
the differential equation. And the same thing for y is equal to zero. That is also a solution to
the differential equation. Now what if we included points, what if we included this point up here, and actually, let me do
it in a different color, so that you could see it, let's say our solution included that point. Well then it might look
something like this, and once again, I'm just
using the slope field as a guide to give me an
idea of what the slope might be as my curve progresses,
as my solution progresses. So solution that includes
the point zero five, might look something like this, and once again, it's just another clue. A solution that includes the
point zero negative 1 1/2, might look something like... might look something like this. So anyway, hopefully this
gives you a better appreciation for why slope fields are interesting. If you have a differential equation that just involves the first derivative, and some x's and y's,
this one only involves the first derivative and y's. We can plot a slope field like this, not too much trouble if we
essentially just keep solving for the slopes, and then
we can use that slope field to get a conceptual or
visual understanding of what the solutions might look like given points that the solutions
might actually contain.