Solved: (a) Verify that y tan (x c) is a one-parameter family of solutions of the

Chapter 1, Problem 30

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(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 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). 31.