(Solution) - Explore the direction fields of the DEs y y2 y -(2025 Original AI-Free Solution)

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Explore the direction fields of the DEs
y' = y2, y' = (y + l )2 , and y' = y2 + 1 .
Describe their similarities and differences. Then answer the following questions:
(a) Suppose each equation has initial condition y(0) = I 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) = l ?
Suppose y = c is an equilibrium or constant solution of the 6nt-onler DE y' = f(y). its basin of a is the set of initial conditions (t, y0) for which solutions tend-to c as t ? ?.