In Problems 21 and 22, first use the method of Example 2 to construct a slope field
Question:
In Problems 21 and 22, first use the method of Example 2 to construct a slope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition. Finally, use this solution curve to estimate the desired value of the solution y(x).
y' = y - x, y(4) = 0; y(-4) = ?
Example 2
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4 3 2 1 20. -t F-1 1-6 VAL -11-1-₁ 11 Example 2 3-1 -1-74 4-Y3 AXAS streto ar Dr N/N RE 10 3 1 33 MIS SE Pir T 3605 MA 13 13 ANN 13.35 -2----- -3 dites. 3\^-^ Solution Figure 1.3.3 shows a table of slopes for the given equation. The numerical slope m = x - y appears at the intersection of the horizontal x-row and the vertical y-column of the table. If you inspect the pattern of upper-left to lower-right diagonals in this table, you can see that it was easily and quickly constructed. (Of course, a more complicated function f(x, y) on the right-hand side of the differential equation would necessitate more complicated calculations.) Figure 1.3.4 shows the corresponding slope field, and Fig. 1.3.5 shows an approximate so- lution curve sketched through the point (-4,4) so as to follow this slope field as closely as possible. At each point it appears to proceed in the direction indicated by the nearby line segments of the slope field. I 2 SP-1 J PIPI (10) 1-4-.. -//![:// " [21 1.17 /1+ 41.4-164-6 -4-3-2-1 0 1 2 3 4 X Construct a slope field for the differential equation y'=x-y and use it to sketch an approx- imate solution curve that passes through the point (-4,4). Although a spreadsheet program (for instance) readily constructs a table of slopes as in Fig. 1.3.3, it can be quite tedious to plot by hand a sufficient number of slope segments as in Fig. 1.3.4. However, most computer algebra systems in- clude commands for quick and ready construction of slope fields with as many line segments as desired; such commands are illustrated in the application material for this section. The more line segments are constructed, the more accurately solution curves can be visualized and sketched. Figure 1.3.6 shows a "finer" slope field for 5 5 FIGURE 1.3.6. Slope field and typical solution curves for y'= x - y. > 0 0 x - 5 FIGURE 1.3.4. Slope field for y = x - y corresponding to the table of slopes in Fig. 1.3.3. 2 -3 dv dt -/ 0 X FIGURE 1.3.5. The solution curve through (-4,4). the differential equation y' = x - y of Example 2, together with typical solution curves treading through this slope field. If you look closely at Fig. 1.3.6, you may spot a solution curve that appears to be a straight line! Indeed, you can verify that the linear function y = x - 1 is a solution of the equation y' = x - y, and it appears likely that the other solution curves approach this straight line as an asymptote as x→ +∞o. This inference illustrates the fact that a slope field can suggest tangible information about solutions that is not at all evident from the differential equation itself. Can you, by tracing the appropriate solution curve in this figure, infer that y(3) 2 for the solution y(x) of the initial value problem y'= x - y, y(-4)= 4? 5 Applications of Slope Fields The next two examples illustrate the use of slope fields to glean useful information in physical situations that are modeled by differential equations. Example 3 is based on the fact that a baseball moving through the air at a moderate speed v (less than about 300 ft/s) encounters air resistance that is approximately proportional to v. If the baseball is thrown straight downward from the top of a tall building or from a hovering helicopter, then it experiences both the downward acceleration of gravity and an upward acceleration of air resistance. If the y-axis is directed downward, then the ball's velocity v = dy/dt and its gravitational acceleration g = 32 ft/s² are both positive, while its acceleration due to air resistance is negative. Hence its total acceleration is of the form =g-kv. A typical value of the air resistance proportionality constant might be k = 0.16. (3)
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Related Book For
Differential Equations And Linear Algebra
ISBN: 9780134497181
4th Edition
Authors: C. Edwards, David Penney, David Calvis
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