Question

For a first-order DEa curve in the plane defined byis called a nullcline of the equation since a lineal element at a point on the curve has zero slopes. Use computer software to obtain a direction field over a rectangular grid of points for, and then superimpose the graph of the nullclineover the direction field. Discuss the behavior of solution curves in regions of the plane defined byand by.

Sketch some approximate solution curves. Try to generalize your observations.

Short answer

Answer:

The slope is zero on the nullclines.

Step-by-step solution

Direction field.

If we systematically evaluateover a rectangular grid of points in the-plane and draw a line element at each pointof the grid with a slopethen the collection of all these line elements is called a direction field or a slope field of the differential equation.

Graph of the direction field.

First, let’s obtain the direction field and then graph the nullclineon it.

Graph of the nullcline.

Let’s also sketch a few solutions. Solutions that start above the nullcline first decrease until they hit the nullcline and then they start to increase. Solutions that start below the nullcline either just increase or they increase until they hit the nullcline, then they decrease until they hit the nullcline again and then they increase.

Hence, the slope is zero on the nullcilnes.