Question

(a) Identify the nullclines (see Problem 17) in Problems 1,3,and 4. With a colored pencil, circle any lineal elements in Figures 2.1.12, 2.1.14, and 2.1.15 that you think maybe a lineal element at a point on a nullcline.

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

Short answer

Answer:

a). For problem 1- they are linesand, for problem 2 nullclines are given as, for problem 3- nullclines areOr.

b). At the point of nullclines, elements are horizontal.

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.

Part a) problem 1.

Let's circle the horizontal elements that are where the null lines pass.

For the first problem, they are given by,

So they are linesand.

Part a)-problem 2.

For the second problem, they are given by,

On our sketch, nullclines are given as a graph of.

Part a)-problem 3.

The third problem is where it gets interesting nullclines to form a sort of net and are given by,

Or.

Solution for part b).

If the equation is autonomous, then it can be written asso the nullclines are given as solutions of. The solutions contain only the variableand can be written asfor a constant. Therefore, the null lines are just horizontal lines.

At the point of nullclines, elements are horizontal.