(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 ϑf/ϑy = 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).