(a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y ´ = 1 + y2.
(b) Since f (x, y) = 1 + y2 and df/dy = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem y ´ = 1 + y2, y(0) = 0. Even though x0 = 0 is in the interval (-2, 2), explain why the solution is not defined on this interval.
(c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b).