Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.
d. Complete the following sentence. The solution of the differential equation with the initial condition y(0)=A, where A is a real number, approaches the line _____ as t→∞.

Verified step by step guidance

1. Identify the given differential equation: \(y'(t) = t - y\).
2. To find the behavior of solutions as \(t \to \infty\), consider the equilibrium or steady-state solution where \(y'(t) = 0\). Set \$0 = t - y\(, which implies \)y = t$.
3. This suggests that as \(t\) becomes very large, the solution \(y(t)\) approaches the line \(y = t\).
4. To confirm this, rewrite the differential equation as \(y' + y = t\), which is a linear first-order ODE. The general solution will involve a particular solution plus a complementary solution that decays over time.
5. Since the complementary solution tends to zero as \(t \to \infty\), the solution \(y(t)\) approaches the particular solution \(y = t\), meaning the solution approaches the line \(y = t\) as \(t \to \infty\).

Key Concepts

Direction Fields

A direction field is a graphical representation of a first-order differential equation showing slope vectors at various points. It helps visualize the behavior of solutions without solving the equation explicitly. Each small line segment indicates the slope y' at that (t, y) point, guiding the shape of solution curves.

Solving First-Order Linear Differential Equations

The equation y' = t - y is a first-order linear differential equation. Solutions can be found using integrating factors or recognizing it as a linear ODE. Understanding the general solution form helps predict long-term behavior and how initial conditions affect the solution.

Asymptotic Behavior and Equilibrium Solutions

As t approaches infinity, solutions to differential equations often approach a particular function or line called an asymptote or equilibrium solution. For y' = t - y, the solution tends to a line where the slope stabilizes, which can be found by setting y' = 0 or analyzing the steady-state behavior.