4 Slope Fields Slope Fields We know that

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4. Slope Fields

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 = 3 x 3+4 x or f’(x)=sin(x) or h”(t)=5 t �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

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 = x 2, if we replace the dy/dx in this equation with what it represents we get slope at any point (x, y) = x 2

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

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 1 2 3 0 0 1 0 1 1 2 2 2 0 4 -1 0 -2 -2 0 -4

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 grapher �http: //mathplotter. lawrenceville. org/mathplotter/m ath. Page/slope. Field. htm? input. Field=2 x+-+y

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Example 6