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


b. In what regions are solutions increasing? Decreasing?


y'(t) = (y−1)(1+y)

Verified step by step guidance

1. Identify the differential equation given: \(y'(t) = (y - 1)(1 + y)\).
2. Determine where the derivative \(y'(t)\) is positive or negative by analyzing the sign of each factor: \((y - 1)\) and \((1 + y)\).
3. Find the critical points where \(y'(t) = 0\), which occur when either \((y - 1) = 0\) or \((1 + y) = 0\). These points are \(y = 1\) and \(y = -1\).
4. Divide the \(y\)-axis into intervals based on these critical points: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\).
5. Test the sign of \(y'(t)\) in each interval by choosing a test value from each interval and substituting it into \(y'(t) = (y - 1)(1 + y)\) to determine where the solutions are increasing (\(y'(t) > 0\)) or decreasing (\(y'(t) < 0\)).

Key Concepts

Differential Equations and Their Solutions

A differential equation relates a function to its derivatives, describing how the function changes. Solutions to differential equations are functions that satisfy this relationship. Understanding the behavior of solutions, such as whether they increase or decrease, involves analyzing the sign of the derivative.

Sign of the Derivative and Monotonicity

The sign of the derivative y'(t) determines whether the solution y(t) is increasing or decreasing. If y'(t) > 0, the function is increasing; if y'(t) < 0, it is decreasing. Identifying intervals where the derivative changes sign helps locate regions of increase or decrease.

Critical Points and Equilibrium Solutions

Critical points occur where y'(t) = 0, indicating potential equilibrium solutions where the function remains constant. For y'(t) = (y−1)(1+y), the critical points are y = 1 and y = -1. These points divide the y-axis into regions that determine the behavior of solutions around them.