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. $$ y^{\prime}=1+2 y $$

Short answer

The behavior of \(y\) as \(t\rightarrow\infty\) depends on the initial value of \(y\). If the initial value \(y(0)\) is non-negative, \(y\) will increase without bound as \(t\rightarrow\infty\). If the initial value \(y(0)\) is negative, \(y\) will approach a limit from below as \(t\rightarrow\infty\).

Step-by-step solution

Calculate the slopes

First, we need to calculate the slope of the tangent line at each point, using the differential equation \(y' = 1+2y\). For example, at the point \((t, y) = (0, 0)\), the slope \(y'\) is given by: $$ y'=1+2(0)=1 $$ At the point \((t, y) = (1, 0)\), the slope \(y'\) equals: $$ y'=1+2(0)=1 $$ You get the idea. We need to do this for multiple points on the grid.

Draw the direction field

Use the calculated slopes to draw arrows at each grid point, which represent the tangents of the solution curves. These arrows make up the direction field. The longer the arrows, the steeper the slope. This visualization will help us analyze the behavior of solutions.

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

Once the direction field is drawn, observe its overall appearance, and analyze the long-term behavior of the solutions, as \(t\) goes to infinity. Notice that the arrows point upward as \(y\) increases, indicating that higher values of \(y\) result in even larger slopes \(y'\), which means the solutions will become steeper over time. Thus, for any non-negative initial value of \(y\), the behavior of \(y\) as \(t\rightarrow\infty\) is that it keeps increasing. For a negative initial value of \(y\), the arrows will point downward, indicating a negative slope, which means the solutions will become less steep. Therefore, for negative initial values of \(y\), the behavior of \(y\) as \(t\rightarrow\infty\) is that it will approach some limit from below. So, the behavior of \(y\) as \(t\rightarrow\infty\) depends on the initial value of \(y\). If \(y(0)\) is non-negative, \(y\) will increase without bound as \(t\rightarrow\infty\). If \(y(0)\) is negative, \(y\) will approach a limit from below as \(t\rightarrow\infty\).

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. Essentially, it tells you how a certain quantity changes over time. In this case, we are working with the differential equation \(y' = 1 + 2y\). This equation relates the derivative \(y'\) of the function \(y\) with the variable itself. What it's saying, in simpler terms, is that for any given value of \(y\), the rate of change of \(y\) is \(1 + 2y\). This differential equation is first-order, meaning it includes first derivatives only. Understanding differential equations is crucial as they model many phenomena, such as changes in population, temperature, or motion, allowing us to predict future behavior.

Slope Analysis

Slope analysis involves calculating the slopes of a given differential equation at various points to understand how the function behaves locally. For the differential equation \(y' = 1 + 2y\), the slope \(y'\) of the tangent line can be calculated at different points. If we take the point \((t, y) = (0, 0)\), the slope is \(1 + 2(0) = 1\). This process is repeated for many points, allowing us to see how the function's behavior changes. A direction field consists of these slopes plotted as small arrows in a grid. These arrows help visualize the solution curves of the differential equation. The longer and steeper arrows indicate greater changes in \(y\), helping students anticipate the trajectory of potential solutions.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input approaches a certain value, commonly towards infinity. By examining the direction field of our differential equation, \(y' = 1 + 2y\), we can infer the asymptotic behavior of solutions. As \(t\) approaches infinity, the general trend is either an increase or leveling off of the function. For our equation, positive initial values of \(y\) result in the solution growing continually larger, showing an upward trend. Conversely, negative initial values of \(y\) show the solutions moving toward a horizontal line, suggesting a plateauing effect. Understanding asymptotic behavior is vital because it predicts the long-term behavior of solutions, which in turn helps in fields like engineering and physics.

Initial Value Problem

An initial value problem is defined by a differential equation along with a specified value at the starting point (\(t=0\)). This boundary condition helps determine a specific solution among many possible ones. For the equation \(y' = 1 + 2y\), the initial value \(y(0)\) affects the behavior of \(y\) over time. If \(y(0)\) is non-negative, \(y\) increases indefinitely as \(t\) grows. However, if \(y(0)\) is negative, \(y\) tends to stabilize or plateau as \(t\) approaches infinity. Initial value problems are crucial in real-world applications where starting conditions influence the projectory, such as in modeling population growth or predicting weather patterns.