Suppose the model in Problem 40 is modi­ed so that air resistance is proportional to v², that is,

M dv/dt = mg - kv².

See Problem 17 in Exercises 1.3. Use a phase portrait to fi­nd the terminal velocity of the body. Explain your reasoning.

Data from problem 17

For a fi­rst-order DE dy/dx = f (x, y) a curve in the plane defi­ned by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction ­field over a rectangular grid of points for dy/dx = x² - 2y, and then superimpose the graph of the nullcline y = ½ x² over the direction fi­eld. Discuss the behavior of solution curves in regions of the plane de­ned by y < ½ x² and by y > ½ x². Sketch some approximate solution curves. Try to generalize your observations.

Data from problem 40

Terminal Velocity In Section 1.3 we saw that the autonomous differential equation

M dv/dt = mg - kv,

where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term 2kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to fi­nd the limiting, or terminal, velocity of the body. Explain your reasoning.