Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.
d. For what values of A are the corresponding solutions decreasing, for t≥0

Verified step by step guidance

1. Identify the differential equation given: \(y' = y(2 - y)\). This represents the slope of the solution curves at any point \((t, y)\).
2. Recall that the solution is decreasing where the derivative \(y'\) is negative. So, we need to find where \(y(2 - y) < 0\).
3. Analyze the inequality \(y(2 - y) < 0\). This product is negative when one factor is positive and the other is negative. So, either \(y < 0\) and \$2 - y > 0\(, or \)y > 0\( and \)2 - y < 0$.
4. Simplify the conditions: For \(y < 0\), \$2 - y\( is always positive, so \)y' < 0\( for all \)y < 0\(. For \)y > 0\(, \)2 - y < 0\( means \)y > 2\(. So, \)y' < 0\( when \)y > 2$.
5. Conclude that the solutions are decreasing for \(t \geq 0\) if the initial condition \(A = y(0)\) satisfies \(A < 0\) or \(A > 2\).

Key Concepts

Direction Fields and Slope Interpretation

A direction field visualizes the slopes of solutions to a differential equation at various points. Each small line segment represents the slope y' at that (t, y) coordinate, indicating the behavior of solution curves without solving the equation explicitly.

Equilibrium Solutions and Stability

Equilibrium solutions occur where y' = 0, meaning the solution is constant. For y' = y(2 - y), equilibria are at y = 0 and y = 2. Stability depends on the sign of y' near these points, determining if solutions approach or move away from equilibria.

Monotonicity of Solutions Based on Initial Conditions

The sign of y' determines whether solutions increase or decrease. For initial value y(0) = A, if y' < 0 for t ≥ 0, the solution decreases. Analyzing y' = y(2 - y) shows that solutions decrease when A > 2 or A < 0, as the product y(2 - y) is negative in these intervals.