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(y - 3)(y + 2)

Verified step by step guidance

1. Identify the given autonomous differential equation: \(y'(t) = y(y - 3)(y + 2)\).
2. Recall that equilibrium solutions occur where the derivative \(y'(t)\) is zero for all \(t\), meaning \(f(y) = 0\).
3. Set the right-hand side equal to zero: \(y(y - 3)(y + 2) = 0\).
4. Solve the equation \(y(y - 3)(y + 2) = 0\) by finding the values of \(y\) that make each factor zero.
5. The solutions to this equation give the equilibrium solutions \(y = 0\), \(y = 3\), and \(y = -2\).

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 use of phase line methods.

Equilibrium Solutions

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

Factoring and Solving Polynomial Equations

To find equilibrium points in equations like y' = y(y - 3)(y + 2), factor the right-hand side and set each factor equal to zero. Solving these polynomial equations yields the constant values of y where the system is in equilibrium.