1. Slope Fields
If you enjoyed connecting the dots, you’ll love slope fields
It is a graphical method to find a particular solution to any differential equation.
a DE is nothing but a slope equation -

2. Slope Fields
A slope field is a graphical picture of a derivative that projects the curve within the picture.
Or a bunch of little line segments that show the slope of the curve (y = ) at different points.

3. Given:
Let’s sketch the slope field …

4. But how?
Substitute the x and y into the differential equation for each of the points.
Plot this slope on the graph.

6. When are the slopes positive?
1st the slopes will be positive when the differential equation is positive, i.e.
And x2 is always positive (except when x=0)
y – 1 > 0 when y > 1
ANSWER: y > 1, but x ≠ 1

7. Given f(0)=3, what is the particular solution.
Separate the variables
Exponentiate both sides
Let K

8. Group Activity Make your own…1 2 4 3

9. What do you notice about the slope fields whose differential equation had only x’s?
What do you notice about the slope fields whose differential equation had only y’s?

10. How to pick out a multiple choice answer (equation) for a slope field…
pay attention to whether you need just the x- or y-values or both
Look for places where the slope is 0
Look at the slopes along the x-axis
(where y = 0)
Look for slopes along the y-axis (where x = 0)
Notice where the slopes are positive and where they are negative

11. Shown is a slope field for which of the following differential equations?B) C) D) E)

12. The thought process… E C A G
Isoclines along horizon. lines → DE depends only on y
Same slope along vert. line → DE depends only on x
Sinusoidal
Consider a specific point – what is the slope there.
What makes the slope zero? What values of x and y make the slope 1?
Note y'(x) = x+y has same slopes along the diagonal.
Solve the separable DE B D I H F K J

13. 6.1 Slope Fields and Euler’s Method,
Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind.
(When this portrait was made he had already lost most of the sight in his right eye.)
Leonhard Euler

14. It was Euler who originated the following notations:
(function notation)
(base of natural log)
(pi)
(summation)
Leonhard Euler
(finite change)

15. 1. Begin at the initial condition. This is the point (x, y).
Instead of finding a particular solution from a slope field, we can approximate a function by starting at a given point and piecing together little segments to build a continuous approximation of the curve.
Euler’s Method for Graphing a Solution to an Initial Value Problem
1. Begin at the initial condition. This is the point (x, y).
2. Find the slope at the initial point.
3. Increase x by a small amount Δx. This is usually given.
Increase y by a small amount Δy. Use Δy = (dy/dx) ∙ Δx.
The new point is (x + Δx, y + Δy).
4. Using the new point, go to step 2 and repeat the process.
5. To construct the graph to the left from the initial point, repeat the process using negative values for Δx.

17. There are many differential equations that can not be solved.
We can still find an approximate solution.
We will practice with an easy one that can be solved.
Initial value:

20. Exact Solution:

21. This is called Euler’s Method.
It is more accurate if a smaller value is used for dx.
It gets less accurate as you move away from the initial value.

22. The book refers to an “Improved Euler’s Method”
The book refers to an “Improved Euler’s Method”. We will not be using it, and you do not need to know it.
The calculator also contains a similar but more complicated (and more accurate) formula called the Runge-Kutta method.
You don’t need to know anything about it other than the fact that it is used more often in real life.
This is the RK solution method on your calculator. p