(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).
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