1. 4. Slope Fields

2. Slope Fields • We know that antidifferentiation, indefinite integration, and solving differential equations all imply the same process • The differential equations we’ve seen so far have been explicit functions of a single variable, like dy/dx = 3x3+4x or f’(x)=sin(x) or h”(t)=5t • Solving these equations meant getting back to y = or f(x)= or h(t)=. • Many times, differential equations are NOT explicit functions of a single variable, and sometimes they are not solvable by analytic methods. • Fear not! There are ways to solve such differential equations. Today we will look at how to solve them graphically.

3. Slope Fields • Slope fields show the general “flow” of a differential equation’s solution. They are an array of small segments which tell the slope of the equation or “tell the equation which direction to go in” • If we have the differential equation dy/dx = x2, if we replace the dy/dx in this equation with what it represents we get slope at any point (x,y) = x2

4. Slope Fields • Consider the following: http://www.hippocampus.org/course_locator;jsessionid=9449EC23D16F51693C3E640FCB76BEB1?course=AP Calculus AB II&lesson=33&topic=2&width=800&height=684&topicTitle=Slope%20Fields&skinPath=http://www.hippocampus.org/hippocampus.skins/default

5. Slope Fields • To construct a slope field, start with a differential equation. We’ll use • Rather than solving the differential equation, we’ll construct a slope field • Pick points in the coordinate plane • Plug in the x and y values • The result is the slope of the tangent line at that point • Draw a small segment at that point with that approximate slope. Make sure your slopes of 0,1,-1 and infinity are correct. All other slopes must be a steepness relative to others around it. • It is impossible to draw a slope field at every point in the x,y plane, so it is restricted to points around the origin

6. Example 1 Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 0 0 0 0 1 0 0 2 0 0 3 0 2 1 0 1 1 2 2 0 4 -1 -2 0 0 -4 -2

7. 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.

8. Example 2 Construct a slope field for y’ = x + y and draw a solution through y(0)=1

9. The more tangent lines we draw, the better the picture of the solutions. There are computer programs and programs for your calculator that will construct them for you. Below is a slope field done on a computer. Notice how we can now, with more confidence and accuracy, draw particular solutions, such as those passing through (0,-2), (0,-1), (0,0), (0,1), and (0,2)

10. Online slope field grapher • http://mathplotter.lawrenceville.org/mathplotter/mathPage/slopeField.htm?inputField=2x+-+y

11. Example 3

12. Example 4

13. Example 5

14. Example 6