1 chapter 7 differential equations: slope fields
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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)