1

Chapter 7Chapter 7

Differential Equations:Differential Equations:Slope Fields

Recall that indefinite integration, or antidifferentiation, is the process of reverting a function from its derivative. In other words, if we have a derivative, the antiderivative allows us to regain the function before it was differentiated – except for the constantexcept for the constant, of course.

If we are given the derivative dy/dx = f ‘(x) and we solve for y (or f (x)), we are said to have found the general solution of a differential equation.

For example: Let

Slope FieldsSlope Fields

,2xdx

dy

dxxdy 2

dxxdy 2then

Cxy 2

And we can easily solve this:This is the general This is the general

solution:solution:

Slope FieldsSlope Fields

We can see that there are several different parabolas that we can sketch in the slope field with varying values of C

When we solve a differential equation this way, we are using an analytical method.

But we could also use a graphically method; the graphical method utilizes slope fieldsslope fields or direction fieldsdirection fields .

Slope fields basically draw the slopes at various coordinates for differing values of C.

For example, the slope field for dy/dx = x is:

x

y

Slope FieldsSlope Fields Let’s examine how we create a slope field. For example, create the slope field for the

differential equation (DE):

y

x

dx

dy

-2

-1

1

2

-2 -1 1 2

x

y

Since dy/dx gives us the slope at any point, we just need to input the coordinate:

At (-2, 2), dy/dx = -2/2 = -1At (-2, 1), dy/dx = -2/1 = -2At (-2, 0), dy/dx = -2/0 = undefinedAnd so on….

This gives us an outline of a hyperbola

x

y

Slope FieldsSlope Fields Let’s examine how we create a slope field. For example, create the slope field for the

differential equation (DE):

y

x

dx

dy

-2

-1

1

2

-2 -1 1 2

x

y

Of course, we can also solve this differential equation analytically:

y

x

dx

dy dxxdyy

dxxdyy

Cxy 22

2

1

2

1

Cxy 22

Cyx 22

Slope FieldsSlope Fields For the given slope field, sketch two

approximate solutions – one of which is passes through the given point:

Now, let’s solve the differential equation passing through the point (4, 2) analytically:

12

1 x

dx

dy dxxdy

1

2

1

dxxdy

1

2

1

C 444

12 2

Cxxy 2

4

1

C2 24

1 2 xxy

Solution:

CC

Slope FieldsSlope Fields

3xdx

dy

In order to determine a slope field from a differential equation, we should consider the following:

i) If isoclinesisoclines (points with the same slope) are along horizontal lines, then DE depends only on y

ii) Do you know a slope at a particular point?

iii) If we have the same slope along vertical lines, then DE depends only on x

iv) Is the slope field sinusoidal?

v) What x and y values make the slope 0, 1, or undefined?

vi) dy/dx = a(x ± y) has similar slopes along a diagonal.

vii) Can you solve the separable DE?

1. _____

2. _____

3. _____

4. _____

5. _____

6. _____

7. _____

8. _____

Match the correct DE with its Match the correct DE with its graph:graph:

2ydx

dy

xdx

dycos

xdx

dysin

yxdx

dy

22 yxdx

dy

1 yydx

dy

y

x

dx

dy

AA BB

CC

EE

GG

DD

FF

HH

HH

BB

FF

DD

GG

EE

AA

Slope FieldsSlope Fields Which of the following graphs could be the

graph of the solution of the differential equation whose slope field is shown?

1998 AP Question: Determine the correct differential equation for the slope field:

Slope FieldsSlope Fields

2 B) xdx

dy

xdx

dy1 A)

y

x

dx

dy D)

yxdx

dy C)

ydx

dyln E)