(a) Identify the nullclines (see Problem 17) in Problems€‘1, 3, and 4. With a colored pencil, circle any lineal elements in figures (1), (2) and (3) that you think may be a lineal element at a point on a nullcline.

(b) What are the nullclines of an autonomous fi­rst-order DE?

(1)

 

(2)

(3)

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 = 1/2 x² over the direction fi­eld. Discuss the behavior of solution curves in regions of the plane defi­ned by y 2 and by y > ½ x². Sketch some approximate solution curves. Try to generalize your observations.

1- 1- 2 -2 -3 1 2 3 -3 -2 -1 3. y 4 -2 -4 -4 -2 2 2.