38–43. 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.
a. Find the equilibrium solutions. 


y′(t) = y(2 - y)

Verified step by step guidance

1. Identify the given autonomous differential equation: \(y'(t) = y(2 - y)\).
2. Recall that equilibrium solutions occur where the derivative \(y'(t)\) is zero, meaning the function \(y(t)\) does not change over time.
3. Set the right-hand side of the differential equation equal to zero to find equilibrium points: \(y(2 - y) = 0\).
4. Solve the equation \(y(2 - y) = 0\) by factoring or using the zero product property, which gives the values of \(y\) where the derivative is zero.
5. The solutions to this equation are the equilibrium solutions \(y = 0\) and \(y = 2\), representing constant solutions where the slope field is horizontal.

Key Concepts

Autonomous Differential Equations

An autonomous differential equation is one where the derivative y' depends only on the variable y, not explicitly on the independent variable t. This means the rate of change of y depends solely on y itself, simplifying analysis and allowing the direction field to be independent of t.

Equilibrium Solutions

Equilibrium solutions occur when y'(t) = 0, meaning the function y(t) remains constant over time. For autonomous equations y' = f(y), equilibrium solutions are found by solving f(y) = 0. These solutions correspond to horizontal lines in the direction field.

Stability of Equilibrium Solutions

Stability refers to whether solutions near an equilibrium tend to move towards or away from it over time. By analyzing the sign of f(y) around equilibrium points, one can determine if the equilibrium is stable (attracting), unstable (repelling), or semi-stable.