Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.
c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

Verified step by step guidance

1. Recall the given differential equation: \(y'(t) = a y + b\), with initial condition \(y(0) = A\), where \(a, b, A \in \mathbb{R}\) and \(a < 0\).
2. Identify the equilibrium solution by setting \(y'(t) = 0\), which gives \(a y + b = 0\). Solve for \(y\) to find the equilibrium point: \(y = -\frac{b}{a}\).
3. Analyze the behavior of the solution relative to the equilibrium \(y = -\frac{b}{a}\). For \(y > -\frac{b}{a}\), substitute a value greater than the equilibrium into \(y'(t) = a y + b\) and use the fact that \(a < 0\) to determine the sign of \(y'(t)\), which indicates whether the solution is increasing or decreasing.
4. Similarly, for \(y < -\frac{b}{a}\), substitute a value less than the equilibrium into \(y'(t)\) and analyze the sign of the derivative to determine if the solution is increasing or decreasing.
5. Use this information to sketch a representative direction field: draw small line segments with slopes given by \(y'(t) = a y + b\) at various points \((t, y)\), showing that solutions above the equilibrium decrease toward it, and solutions below increase toward it, confirming the stability of the equilibrium and the behavior of solutions depending on whether \(A > -\frac{b}{a}\) or \(A < -\frac{b}{a}\).

Key Concepts

Direction Fields (Slope Fields)

A direction field is a graphical tool that represents the slopes of solutions to a differential equation at various points. For y' = ay + b, each point (t, y) has a slope given by ay + b. Plotting small line segments with these slopes helps visualize the behavior of solutions without solving the equation explicitly.

Equilibrium Solutions and Stability

An equilibrium solution occurs where the derivative y' = 0, here at y = -b/a. This constant solution divides the behavior of other solutions. When a < 0, the equilibrium is stable, meaning solutions near it tend to move towards it over time, influencing whether solutions increase or decrease based on their initial value relative to -b/a.

Initial Value Problems and Solution Behavior

An initial value problem specifies a starting point y(0) = A. The sign of a and the position of A relative to the equilibrium -b/a determine the solution's trend. For a < 0, if A > -b/a, the solution decreases toward the equilibrium; if A < -b/a, it increases toward it, reflecting the system's dynamics over t ≥ 0.