(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 ∂f/∂y = 2y are continuous everywhere, the region R 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 xₒ = 0 is in the interval (-2, 2), explain why the solution is not defined on this interval.

1- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis.2- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in the first question?