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Unformatted text preview: 6.4. SLOPE FIELDS AND EULER’S METHOD 783 6.4 Slope Fields and Euler’s Method Not all equations are separable, many separable equations do not lead to explicit solutions, and even when you find a solution it may be so complex that it is nearly impossible to interpret what it means. To address these issues, we discuss a qualitative method, slope fields , and a numerical method, Euler’s method , for studying solutions of differential equations. Slope fields Consider a differential equation of the form dy dt = f ( t,y ) where f ( t,y ) denotes an expression involving t and y . Since a solution y ( t ) to this differential equation satisfies y ′ ( t ) = f ( t,y ( t )), it follows that the slope of all solutions at time t are given by the right hand side f ( t,y ( t )) of the differential equation. Equivalently, a solution through a point ( t,y ) is tangent to a line that passing through the point ( t,y ) and that has slope f ( t,y ). A qualitative way to investigate the behavior of solutions to dy dt = f ( t,y ) is to sketch its slope field . We introduced slope fields in Section 5.1. Recall, a slope fields is a figure in the typlane with infinitesimal line segments of slope f ( t,y ) at ( t,y ). There are two ways to generate slope fields. One method is by using technology and the other is by hand. We will not construct a slope field for dy/dt = 1 /t by hand. Since for t = 1 the slope is 1 1 = 1, we draw short line segments at t = 1, each with slope 1, for different yvalues, as shown in Figure 6.16a. If t = − 3, then the slope is − 1 3 and we draw short line segments at t = − 3, each with slope − 1 / 3, also shown in Figure 6.16a. If we continue to plot these slope points for different values of t , we obtain many little slope lines. The resulting graph, shown in Figure 6.16b is the slope field for the equation dy dt = 1 /t . Finally, notice the relationship between the slope field for dy dt = 1 /t and its solutions y = ln  t  + C . If we choose particular values for C , say C = 0 ,C = − ln2, or C = 2, and draw these particular antiderivatives as shown in Figure 6.16c, we notice that these particular solutions are anticipated by the slope field drawn in part b. While the slope field for dy dt = 1 /t was relatively straightforward to sketch, these sketches can be more challenging for differential equations with a more complicated right hand side. In the next example, we illustrate how to handle these cases. Example 1. Solving a differential equation using a slope field © 2010 Schreiber, Smith & Getz 784 6.4. SLOPE FIELDS AND EULER’S METHOD a. Beginning of a slope field b. slope field c. Particular solutions Figure 6.16: Solution of the differential equation y ′ = 1 y using a slope field Consider a drug that continuously infuses at a periodic rate into a patient. One possible differential equation modeling such a scenario is dy dt = 10 + 10 sin t − y where y is the amount of drug (in mg) and t is the time in units of h (hours). Notice that this is just a mixingis the time in units of h (hours)....
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 Fall '10
 WayneM.Getz
 Numerical Analysis, Slope, slope field, dy dt, Smith & Getz, ©2010 Schreiber

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