Transcription of Chapter 16 F D IRST IFFERENTIAL -ORDER EQUATIONS
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Chapter 16 F D IRST IFFERENTIAL -ORDER EQUATIONS
OVERVIEWIn Section we introduced differential EQUATIONS of the form ,where is given and yis an unknown function of . When is continuous over some inter-val, we found the general solution by integration, . In Section wesolved separable differential EQUATIONS . Such EQUATIONS arise when investigating exponen-tial growth or decay, for example. In this Chapter we study some other types of first-orderdifferential EQUATIONS . They involve only first derivatives of the unknown (x) dx y(x) x dy>dx= (x)16-1 FIRST-ORDERDIFFERENTIALEQUATIONSC hapter16 Solutions, Slope Fields, and Picard s TheoremWe begin this section by defining general differential EQUATIONS involving first then look at slope fields, which give a geometric picture of the solutions to such equa-tions.
tion curves. Figure 16.2a shows a slope field, with a particular solution sketched into it in Figure 16.2b. We see how these line segments indicate the direction the solution curve takes at each point it passes through. sx 0, y 0d ƒsx 0, y 0d y¿=ƒsx, yd, ysx 0d = y 0 ys0d = csx + 1d - 1 3 e x d x=0 = 1 - 1 3 = 2 3. y-x = sx + 1d - 1 3 e x-x ...
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