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Chapter 7

Chapter 7. Differential Equations: Slope Fields . Slope Fields.

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Chapter 7

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  1. Chapter 7 Differential Equations: Slope Fields

  2. Slope Fields • Recall that indefinite integration, or antidifferentiation, is the process of reverting a function from its derivative. In other words, if we have a derivative, the antiderivative allows us to regain the function before it was differentiated – except for the constant, of course. • If we are given the derivative dy/dx = f ‘(x) and we solve for y (or f (x)), we are said to have found the general solution of a differential equation. • For example: Let And we can easily solve this: This is the general solution:

  3. Slope Fields • When we solve a differential equation this way, we are using an analytical method. • But we could also use a graphically method; the graphical method utilizes slope fieldsordirection fields. • Slope fields basically draw the slopes at various coordinates for differing values of C. • For example, the slope field for dy/dx = x is: We can see that there are several different parabolas that we can sketch in the slope field with varying values of C

  4. Slope Fields • Let’s examine how we create a slope field. • For example, create the slope field for the differential equation (DE): Since dy/dx gives us the slope at any point, we just need to input the coordinate: At (-2, 2), dy/dx = -2/2 = -1 At (-2, 1), dy/dx = -2/1 = -2 At (-2, 0), dy/dx = -2/0 = undefined And so on…. This gives us an outline of a hyperbola

  5. Slope Fields • Let’s examine how we create a slope field. • For example, create the slope field for the differential equation (DE): Of course, we can also solve this differential equation analytically:

  6. Solution: Slope Fields • For the given slope field, sketch two approximate solutions – one of which is passes through the given point: Now, let’s solve the differential equation passing through the point (4, 2) analytically:

  7. Slope Fields Match the correct DE with its graph: • In order to determine a slope field from a differential equation, we should consider the following: • If isoclines (points with the same slope) are along horizontal lines, then DE depends only on y • Do you know a slope at a particular point? • If we have the same slope along vertical lines, then DE depends only on x • Is the slope field sinusoidal? • What x and y values make the slope 0, 1, or undefined? • dy/dx = a(x ±y) has similar slopes along a diagonal. • Can you solve the separable DE? A B H 1. _____ F 2. _____ C D D 3. _____ C 4. _____ E F A 5. _____ G 6. _____ G H E 7. _____ B 8. _____

  8. Slope Fields • Which of the following graphs could be the graph of the solution of the differential equation whose slope field is shown?

  9. Slope Fields • 1998 AP Question: Determine the correct differential equation for the slope field:

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