4. Slope Fields

Slope FieldsWe 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.

Slope FieldsSlope 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

Slope FieldsConsider 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

Slope FieldsTo construct a slope field, start with a differential

equation. We’ll use Rather than solving the differential equation, we’ll

construct a slope fieldPick points in the coordinate planePlug in the x and y valuesThe result is the slope of the tangent line at that pointDraw 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

/ 2dy dx x

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 00 1 00 00 0

23

1 0 21 1 2

2 0 4

-1 0 -2

-2 0 -4

Example 1

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.

Example 2

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

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)

Online slope field grapherhttp://mathplotter.lawrenceville.org/mathpl

otter/mathPage/slopeField.htm?inputField=2x+-+y

Example 3

Example 4

Example 5

Example 6