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Transcript of Slope Fields and Euler’s Method Copyright © Cengage Learning. All rights reserved. 6.1 6.1 Day 2...
Slope Fields and Euler’s Method
Copyright © Cengage Learning. All rights reserved.
6.1
6.1 Day 2 2014
4
6.1 Day 2: Slope Fields
Greg Kelly, Hanford High School, Richland, Washington
5
A little review:
Consider:2 3y x
then: 2y x 2y x
2 5y x or
It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.
However, when we try to reverse the operation:
Given: 2y x find y
2y x C
We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.
This is the general solution
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If we have some more information we can find C.
Given: and when , find the equation for .2y x y4y 1x
2y x C 24 1 C
3 C 2 3y x
This is called an initial value problem. We need the initial values to find the constant.
An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.
2y x
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Slope Fields
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Slope Fields
Solving a differential equation analytically can be difficult or even impossible. However, there is a graphical approach you can use to learn a lot about the solution of a differential equation.
Consider a differential equation of the formy' = F(x, y) Differential equation
where F(x, y) is some expression in x and y.
At each point (x, y) in the xy–plane where F is defined, the differential equation determines the slope y' = F(x, y) of the solution at that point.
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Slope FieldsIf you draw short line segments with slope F(x, y) at selected points (x, y) in the domain of F, then these line segments form a slope field, or a direction field, for the differential equation y' = F(x, y).
Each line segment has the same slope as the solution curve through that point.
A slope field shows the general shape of all the solutions and can be helpful in getting a visual perspective of the directions of the solutions of a differential equation. Slope fields are graphical representations of a differential equation which give us an idea of the shape of the solution curves. The solution curves seem to lurk in the slope field.
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Slope Fields
A slope field
shows the
general shape
of all solutions
of a differential
equation.
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Sketching a Slope Field
Sketch a slope field for the differential equationby sketching short segments of the derivative at several points.
2y x
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Draw a segment with slope of 2.
Draw a segment with slope of 0.
Draw a segment with slope of 4.
2y x
x y y0 0 0
0 1 0
0 0
0 0
2
3
1 0 2
1 1 2
2 0 4
-1 0 -2
-2 0 -4
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2y x
If you know an initial condition, such as (1,-2), you can sketch the curve.
By following the slope field, you get a rough picture of what the curve looks like.
In this case, it is a parabola.
Slope fields show the general shape of all solutions of a differential equation.
We can see that there are several different parabolas that we can sketch in the slope field with varying values of C.
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x
y
Slope Fields Create the slope field for the differential
equationy
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
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Given:
2 ( 1)dy
x ydx
Let’s sketch the slope field …
Slope Fields
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2( 1)d
xy
xy
d
Separate the variables
2
1
yd
y
dxx
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1dy x dx
y
31ln 1
3y x C
3 31 13 31 x C x Cy e e e
313
313
1 , apply IC
1
(0) 3
2
x
x
y e
y
fK
e
Given f(0)=3, find the particular solution.
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CC
Slope Fields
3xdx
dy
In order to determine a slope field for a differential equation, we should consider the following:
i) If 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
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Slope Fields
Which of the following graphs could be the graph of the solution of the differential equation whose slope field is shown?
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1998 AP Question: Determine the correct differential equation for the slope field:
Slope Fields
2 B) xdx
dy
xdx
dy1 A)
y
x
dx
dy D)
yxdx
dy C)
ydx
dyln E)
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Homework Slope Fields Worksheet
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Euler’s Method – BC Only
Euler’s Method is a numerical approach to approximating the particular solution of the differential equation
y' = F(x, y)
that passes through the point (x0, y0).
From the given information, you know that the graph of the solution passes through the point (x0, y0) and has a slope of
F(x0, y0) at this point.
This gives you a “starting point” for approximating the solution.
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Euler’s Method
From this starting point, you can proceed in the direction indicated by the slope.
Using a small step h, move along the
tangent line until you arrive at the
point (x1, y1) where
x1 = x0 + h and y1 = y0 + hF(x0, y0) as shown in Figure 6.6.
Figure 6.6
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Euler’s Method
If you think of (x1, y1) as a new starting point, you can
repeat the process to obtain a second point (x2, y2).
The values of xi and yi are as follows.
1 0 0 0 2 1 1 1, , , ,y y deriv x y x y y deriv x y x etc
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Example 6 – Approximating a Solution Using Euler’s Method
Use Euler’s Method to approximate the particular solution of the differential equation
y' = x – y
passing through the point (0, 1). Use a step of h = 0.1.
Solution:Using h = 0.1, x0 = 0, y0 = 1, and F(x, y) = x – y, you have x0 = 0, x1 = 0.1, x2 = 0.2, x3 = 0.3,…, and
y1 = y0 + hF(x0, y0) = 1 + (0 – 1)(0.1) = 0.9
y2 = y1 + hF(x1, y1) = 0.9 + (0.1 – 0.9)(0.1) = 0.82
y3 = y2 + hF(x2, y2) = 0.82 + (0.2 – 0.82)(0.1) = 0.758.
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Example 6 – Solution
Figure 6.7
The first ten approximations are shown in the table.
cont’d
You can plot these values to see a
graph of the approximate solution,
as shown in Figure 6.7.
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Homework Slope Fields Worksheet BC add pg. 411 69-73 odd