1. Differential Equations: Slope Fields
Chapter 7
Differential Equations:
Slope Fields

2. 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 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
And we can easily solve this:
This is the general solution:

3. Slope Fields
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 fields or direction 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:
We can see that there are several different parabolas that we can sketch in the slope field with varying values of C

4. Slope Fields Let’s examine how we create a slope field.
For example, create the slope field for the differential equation (DE):
Since dy/dx gives us the slope at any point, we just need to input the coordinate:
At (-2, 2), dy/dx = -2/2 = -1
At (-2, 1), dy/dx = -2/1 = -2
At (-2, 0), dy/dx = -2/0 = undefined
And so on….
This gives us an outline of a hyperbola

5. Slope Fields Let’s examine how we create a slope field.
For example, create the slope field for the differential equation (DE):
Of course, we can also solve this differential equation analytically:

6. Slope 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:
Solution:

7. Slope Fields
Match the correct DE with its graph:
In order to determine a slope field from a differential equation, we should consider the following:
If isoclines (points with the same slope) are along horizontal lines, then DE depends only on y
Do you know a slope at a particular point?
If we have the same slope along vertical lines, then DE depends only on x
Is the slope field sinusoidal?
What x and y values make the slope 0, 1, or undefined?
dy/dx = a(x ± y) has similar slopes along a diagonal.
Can you solve the separable DE? A B H
1. _____ F
2. _____ C D
3. _____ D C
4. _____ E F A
5. _____ G
6. _____ G H E
7. _____ B
8. _____

8. Slope Fields
Which of the following graphs could be the graph of the solution of the differential equation whose slope field is shown?

9. Slope Fields
1998 AP Question: Determine the correct differential equation for the slope field: