Consider the family of differential equations given by dy dx =y (1- y L ), L>0

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the family of differential equations given by dydx=y(1−yL),L>0.\frac{dy}{dx}=y\Bigl(1-\frac{y}{L}\Bigr), \qquad L>0.The positive constant LL varies and represents a parameter of the family. Answer the following questions about how changes in LL influence the slope field and the qualitative behavior of solutions. (a) Determine all equilibrium solutions in terms of LL and classify each as stable or unstable based on the local slopes in the slope field. (b) Describe how the spacing and steepness of the line elements in the slope field between y=0y=0 and y=Ly=L change as LL increases while the xx–scale is kept fixed. (c) Suppose a solution curve starts at the point (0,12L)(0,\tfrac12L). Explain how the time (measured in the xx–direction) it takes the solution to reach y=34Ly=\tfrac34L depends on LL. Provide reasoning based on the appearance of the slope field rather than an explicit integration. (d) Discuss the limiting form of the slope field as L→∞L\to\infty and interpret what this limit suggests about the corresponding differential equation.

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