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4. Slope Fields

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4. 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.

- 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

- 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

- 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

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

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.

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)

- http://mathplotter.lawrenceville.org/mathplotter/mathPage/slopeField.htm?inputField=2x+-+y

Example 3

Example 4

Example 5

Example 6