17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).


y'(t) = (y−2)(y+1)

Verified step by step guidance

1. Identify the equilibrium solutions by setting the derivative equal to zero: \(y'(t) = 0\).
2. Set the right-hand side of the differential equation equal to zero: \((y - 2)(y + 1) = 0\).
3. Solve the equation \((y - 2)(y + 1) = 0\) for \(y\) to find the constant solutions.
4. The solutions to this equation give the equilibrium solutions where \(y(t)\) does not change over time.
5. Write the equilibrium solutions explicitly as \(y(t) = 2\) and \(y(t) = -1\) for all \(t \geq 0\).

Key Concepts

Equilibrium Solutions of Differential Equations

Equilibrium solutions occur when the derivative y'(t) equals zero for all t, meaning the solution y(t) remains constant over time. To find these, set the right-hand side of the differential equation to zero and solve for y. These solutions represent steady states where the system does not change.

Factoring and Solving Polynomial Equations

To find equilibrium points in y'(t) = (y−2)(y+1), factor the expression and set each factor equal to zero. This yields values of y that make the derivative zero. Understanding how to factor and solve polynomial equations is essential for identifying these constant solutions.

Interpretation of Differential Equations Without Direction Fields

Even without a direction field, analyzing the sign of y'(t) around equilibrium points helps determine whether solutions increase or decrease. This qualitative analysis aids in understanding the behavior of solutions near equilibrium without graphing, focusing on algebraic manipulation and sign analysis.