Chapter D1.2, Problem 41E

Equilibrium solutions A differential equation of the form y′(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y₀ is an equilibrium solution of the equation provided f(y₀) = 0 (because then y′(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

1. a. Find the equilibrium solutions.
2. b. Sketch the direction field, for t ≥ 0.
3. c. Sketch the solution curve that corresponds to the initial condition y (0) = 1.

41.    y′ (t) = y (y – 3)