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Unformatted text preview: (9/8/08) Math 10B. Lecture Examples. Section 11.4. Separation of variables Example 1 Figure 1 shows the slope field of the differential equation dy dx = y and Figure 2 shows the graphs of eight solutions. (a) Use the differential equation to explain the pattern of the slope lines. (b) Find an equation for all solutions. x 1- 1 2- 2 y- 1 1 x 1- 1 2- 2 y- 1 1 FIGURE 1 FIGURE 2 Answer: (a) One description and explanation: The lines in the slope field of dy dx = y in Figure 1 have the same slope along each horizontal line because the formula on the right does not involve x . The lines are horizontal along the x-axis where y = 0, have positive slopes above the x-axis where y > 0, and have negative slopes below the x-axis where y < 0, and they become steeper as y increases through positive values or decreases through negative values. (b) The solutions are y = Ce x with arbitrary constants C ....
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