A differential equation of the form dx/dt = f(x), whose right-hand side does not explicitly depend on the independent variable t, is called an autonomous differential equation. For example, the logistic model in Example 5 was autonomous. For the autonomous differential equations in Exercise, perform each of the following tasks. Note that the first three tasks are to be performed without the aid of technology.
a) Set the right-hand side of the differential equation equal to zero and solve for the equilibrium points.
b) Plot the graph of the right-hand side of each autonomous differential equation versus x, as in Figure 3.16. Draw the phase line below the graph and indicate where x is increasing or decreasing, as was done in Figure 3.16.
c) Use the information in parts a) and b) to draw sample solutions in the xt plane. Be sure to include the equilibrium solutions.
d) Check your results with dfield6. Again, be sure to include the equilibrium solutions.
e) If x0 is an equilibrium point, i.e., if f(x0) = 0, then x(t) = x0 is an equilibrium solution. It can be shown that if f′(x0)<0, then every solution curve that has an initial value near x0 converges to x0 as t → ∞. In this case x0 is called a stable equilibrium point. If f′(x0) > 0, then every solution curve that has an initial value near x0 diverges away from x0 as t → ∞, and x0 is called an unstable equilibrium point. If f′(x0) = 0, no conclusion can be drawn about the behavior of solution curves. In this case the equilibrium point may fail to be either stable or unstable. Apply this test to each of the equilibrium points.
x′ = cos(πx), x ∈ [−3,3].
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