Consider the initial-value problem: (a) Verify that the hypotheses of the e...

Consider the initial-value problem:

(a) Verify that the hypotheses of the existence and uniqueness theorem are satisfied for this initialvalue problem for any x₀, y₀. This establishes that the initial-value problem always has a unique solution on some interval containing x₀.

(b) By inspection, determine all equilibrium solutions to the differential equation.

(c) Determine the regions in the xy-plane where the solution curves are concave up, and determine those regions where they are concave down.

(d) Sketch the slope field for the differential equation, and determine all values of y0 for which the initial-value problem has bounded solutions. Onyour slope field, sketch representative solution curves in the three cases y₀ < 0, 0 < y₀ < 1, and y₀ > 1.