Chapter 1, Problem 18RE

1. (a) Verify that the one-parameter family y² − 2y = x² − x + c is an implicit solution of the differential equation

   (2y − 2)y′ = 2x − 1.

2. (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) = 1.
3. (c) Use your result in part (b) to find an explicit function y = ϕ(x) that satisfies y(0) = 1. Give the domain of the function ϕ. Is y = ϕ(x) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.