Consider the initial value problem

Chapter 1, Problem 1.109

(choose chapter or problem)

Consider the initial value problem |y | = 2y; y(x0) = y0, in which x0 is any number. (a) Assuming that y0 > 0, find two solutions. (b) Explain why the conclusion of part (a) does not violate Theorem 1.2. Theorem 1.2 can be proved using Picard iterates. Here is the idea. Consider the initial value problem y = f (x, y); y(x0) = y0. For each positive integer n, define yn (x) = y0 + x x0 f (t, yn1(t)) dt. T This is a recursive definition, giving y1(x) in terms of y0, then y2(x) in terms of y1(x), and so on. The functions yn (x) are called Picard iterates for the initial value problem. Under the assumptions of the theorem, the sequence of functions yn (x) converges for all x in some interval about x0, and the limit of this sequence is the solution of the initial value problem on this interval. In each of 6 through 9: (a) Use Theorem 1.2 to show that the problem has a solution in some interval about x0. (b) Find this solution. (c) Compute Picard iterates y1(x) through y6(x), and from these, guess yn (x) in general. (d) Find the Taylor series of the solution from part (b) about x0.