(a) Verify that is a one-parameter family of solutions of the differential equation...

(a) Verify that is a one-parameter family of solutions of the differential equation

(b) Since f (x, y) = y² and are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0) = 1. Then find a solution from the family in part (a) that satisfies y(0)= −1. Determine the largest interval I of definition for the solution of each initial value problem.

(c) Determine the largest interval I of definition for the solution of the first-order initial-value problem [Hint: The solution is not a member of the family of solutions in part (a).]