Every autonomous first-order equation dy/dx f (y) is separable. Find explicit solutions

Chapter 2, Problem 43

(choose chapter or problem)

Every autonomous first-order equation dy/dx = f (y) is separable. Find explicit solutions \(y_{1}(x), y_{2}(x), y_{3}(x), \text { and } y_{4}(x)\) of the differential equation \(d y / d x=y-y^{3}\) that satisfy, in turn, the initial conditions \(y_{1}(0)=2\), \(y_{2}(0)=\frac{1}{2}\), \(y_{3}(0)=-\frac{1}{2}\), and \(y_{4}(0)=-2\). Use a graphing utility to plot the graphs of each solution. Compare these graphs with those predicted in Problem 19 of Exercises 2.1. Give the exact interval of definition for each solution.