Comparison Explore the direction fields of the DEsDescribe their similarities and differen...

Comparison Explore the direction fields of the DEs

Describe their similarities and differences. Then answer the following questions:

(a) Suppose each equation has initial condition y(0) = 1. Is one solution larger than the other for t > 0?

(b) You can verify that y = 1/(1 − t) satisfies the IVP y' = y2, y(0) = 1. What does this say about the solution of y' = y2 + 1, y(0) = 1?

Suppose y ≡ c is an equilibrium or constant solution of the first-order DE y' = f(y). Its basin of attraction is the set of initial conditions (t, y0) for which solutions tend to c as t → ∞.

An example of shading a basin is shown in Fig. 9.

Figure 9 For y' = y2 − 4, the shaded area is the basin of attraction for the stable equilibrium solution y = −2.