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. $$ y^{\prime}=e^{-t}+y $$

Short answer

Answer: If the initial value of \(y\) is greater than -1, then \(y\) increases without bound as \(t \rightarrow \infty\). If the initial value is less than or equal to -1, \(y\) approaches -1 as \(t \rightarrow \infty\).

Step-by-step solution

Set up a coordinate system

Create a coordinate system with the horizontal axis representing time (\(t\)) and the vertical axis representing the dependent variable (\(y\)). It's helpful to choose an appropriate range for the axes, such as \(-1 \leq t\leq 4\) and \(-3 \leq y \leq 3\).

Compute the slope

For any point \((t, y)\) on the graph, the slope of the tangent line to the solution curve of the differential equation is given by \(y'=e^{-t}+y\). Calculate the slope at multiple points (e.g., grid points) on the coordinate system.

Draw the direction field

At each computed grid point, draw a small tangent line that represents the slope \(y'=e^{-t}+y\). These tangent lines give us a visual representation of the direction field for our differential equation.

Sketch solution curves

By looking at the direction field, we can now draw a few solution curves that follow the tangent lines, representing the trajectory of the solutions for different initial values at \(t=0\).

Analyze the behavior of \(y\) as \(t\rightarrow \infty\)

Observe the solution curves for various initial values and their behavior as \(t\) approaches infinity. As the solution curves separate from the horizontal axis, it is evident that \(y\) increases without bound as \(t \rightarrow \infty\). Moreover, as we look at solution curves that start below the equilibrium line (y = -1), we see that they approach an asymptote around y = -1 as\( t \rightarrow \infty\). Thus, the behavior of \(y\) depends on the initial value of \(y\). In conclusion, for the given differential equation \(y^{\prime}=e^{-t}+y\), the behavior of \(y\) as \(t \rightarrow \infty\) depends on the initial value of \(y\) at \(t=0\). If the initial value of \(y\) is greater than -1, then \(y\) increases without bound as \(t \rightarrow \infty,\) otherwise it approaches -1.

Key concepts

Differential Equation

A differential equation is a mathematical equation that involves an unknown function and its derivatives. In simpler terms, it's an equation that shows the relationship between a function and the rates at which it changes. The key part of a differential equation is the derivative, like in our example where we see it as \(y' = e^{-t} + y\). Here, \(y'\) indicates the derivative of \(y\) with respect to \(t\), which essentially tells us how \(y\) changes when \(t\) changes.

Differential equations are widely used because they can describe a variety of dynamic systems, from population growth to the motion of objects. They can be first-order, like our example, where only the first derivative appears, or higher-order, if more complex derivatives are involved. They are essential tools in physics, engineering, biology, and many other fields where one needs to model continuous systems.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation that comes with an additional piece of information: the value of the unknown function at a particular point. It tells us the starting condition of the problem and allows us to find a specific solution. Think of it as knowing the starting point on a map, helping us to find our way to the destination.

In the context of our exercise, the initial value problem involves determining the behavior of \(y\) given its value at \(t=0\). This is crucial because the behavior of the solution paths in the direction field can change based on this initial condition. Essentially, when you change the starting value of \(y\), it influences the path that the function will take over time.

Equilibrium Solution

An equilibrium solution is a stable solution to a differential equation where the derivative equals zero. This means that once the system reaches this state, it remains unchanged over time because there is no net change occurring. In our case, we find equilibrium when \(e^{-t} + y = 0\), which suggests that these points do not evolve as time progresses.

In the given example, the equilibrium occurs around \(y = -1\). When the solutions start below this line, they tend to approach it as \(t\) increases, demonstrating that this is a stable equilibrium. On a direction field, equilibrium solutions often appear as lines or curves where all the direction vectors are horizontal, emphasizing the lack of movement.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input becomes very large or very small. This usually involves describing the function's tendency towards certain values or directions. Essentially, it's like describing the long-term trends of a system.

In the context of our differential equation, as \(t\) approaches infinity, examining the asymptotic behavior helps us understand how the solution behaves far into the future. Some solution curves might rise indefinitely if they start above the equilibrium level, signifying an unbounded increase. If they start below, they may approach and eventually flatten out at the equilibrium point \(y = -1\). These patterns lend crucial insights into the stability and long-term predictions for systems modeled by differential equations.