Question

draw a direction field for the given differential equation. Based on the direction field, determine the behavior of \(y\) as \(t \rightarrow \infty\). If this behavior depends on the initial value of \(y\) at \(t=0,\) describe this dependency. Note the right sides of these equations depend on \(t\) as well as \(y\), therefore their solutions can exhibit more complicated behavior than those in the text. $$ y^{\prime}=-2+t-y $$

Short answer

Answer: As t approaches infinity, y also approaches infinity. The behavior of y depends on its initial value at t=0. For smaller initial values of y at t=0, the solution trajectories grow slower as compared to larger initial values. However, in both cases, as t approaches infinity, y approaches infinity.

Step-by-step solution

(Step 1: Drawing the Direction Field)

To draw the direction field for the given differential equation \(y' = -2 + t - y\), we need to plot the vector field resulting from evaluating the equation at different points in the \(t-y\) plane. The right side of the given differential equation represents the slope of the tangent at every point \((t,y)\) in the plane. So, we will plot a small line segment (tangent) at different points in the plane whose slope is given by the right side of the differential equation.

(Step 2: Determine the Behavior of y as t approaches infinity)

To determine the behavior of y as t approaches infinity, we will analyze the direction field we have plotted in step 1. By observing the pattern of the tangent lines in the direction field, we can predict the possible trajectories of the solutions. As t grows, we can notice that the tangent lines are getting steeper with positive slopes. This indicates that the solution \(y(t)\) will grow with t, and as \(t \rightarrow \infty\), \(y \rightarrow \infty\).

(Step 3: Describe the Dependency on Initial Values of y at t=0)

Finally, we can identify the dependency of the behavior of y on its initial value at t=0. By observing the direction field, we can see that for smaller initial values of y at t=0, the solution trajectories grow slower than for larger initial values. In conclusion, the behavior of \(y\) as \(t \rightarrow \infty\) does depend on the initial value of \(y\) at \(t=0\). For smaller initial values of \(y\) at t=0, the solution trajectories grow slower as compared to larger initial values. In both cases, however, as \(t \rightarrow \infty\), \(y \rightarrow \infty\).

Key concepts

Slope Fields

Understanding slope fields, which are also known as direction fields, is essential when working with differential equations. They provide a visual representation of all possible solutions to a differential equation. To create a slope field, we draw small line segments or arrows, which we call 'tangents', that represent the slope given by a differential equation at various points.

For the differential equation in our exercise, given by \(y' = -2 + t - y\)\, imagine plotting the equation's right-hand side value for \(y\) and various points for \(t\) on a plane. The collection of these tangents forms the slope field. This helps us visualize how solutions might look without solving the equation analytically. The steeper the slope, the greater the rate of change at that particular point.

Asymptotic Behavior

Asymptotic behavior refers to the tendency of a function or sequence to approach a particular line or value as the input grows infinitely large or small. In the context of differential equations, analyzing the direction or slope field can illuminate a function's long-term behavior — namely, how the dependent variable \(y\) behaves as independent variable \(t\) moves towards infinity.

In our example, by examining the pattern in the slope field, we surmise that as \(t\) increases, the function \(y(t)\) appears to increase indefinitely. This implies that the solution curve will not level off or approach a horizontal asymptote but will instead grow steeper and tend to infinity with \(t\), suggesting an asymptotic behavior where \(y\) does not settle toward a finite value.

Initial Value Problems

An initial value problem is a specific type of differential equation accompanied by a condition that specifies the value of the unknown function at a given point. It is an important concept that allows us to find a unique solution to a differential equation. By knowing the initial state of the variable \(y\) at the starting point \(t=0\), we can determine which solution curve from the slope field matches this condition.

In the exercise provided, the dependency of the function \(y(t)\) on the initial value suggests that the starting point \(y(0)\) affects how quickly the function grows. Solutions with a larger initial value of \(y\) will have steeper slopes initially, which indicates a faster growth as \(t\) increases, compared to those with a smaller initial value.

Tangent Lines in Differential Equations

The concept of tangent lines in differential equations is related to the graphical approach of analyzing slope fields. Each line segment in a slope field represents the tangent to a solution curve at that point. The slope of the tangent line at any point \( (t, y) \) is given by the differential equation itself, \(y'\), at that point.

The equation \(y' = -2 + t - y\) from our problem gives us the slope of the tangent to the curve at any point. So, when we sketch these tangent lines across the plane for various \(t\) and \(y\) values, we are actually constructing a map of instantaneous rates of change. This serves as a detailed guide to understanding the dynamic nature of the solutions to the differential equation.