A. verify that y = tan(x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y2.

1. differentiating y = tan(x + c) we get y' = sec(x + c) or y' = 1 + y2.
2. differentiating y = tan(x + c) we get y' = 1 + tan2(x + c) or y' = 1 + y2.
3. differentiating y = tan(x + c) we get y' = csc(x + c) or y' = 1 + y2.
4. differentiating y = tan(x + c) we get y' = tan2(x + c) or y' = 1 + y2.
5. differentiating y = tan(x + c) we get y' = 1 + sec2(x + c) or y' = 1 + y2.

b. since f(x, y) = 1 + y2 and âf/ây = 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.

y =

even though x0 = 0 is in the interval (â2, 2), explain why the solution is not defined on this interval. since tan(x) is discontinuous at x = â± , the solution is not defined on (â2, 2).
c. determine the largest interval i of definition for the solution of the initial-value problem in part (b). (enter your answer using interval notation.)