Question

Slope Field Use the slope field for the differential equation \(y^{\prime}=1 / y,\) where \(y>0\) , to sketch the graph of the solution that satisfies each given initial condition. Then make a conjecture about the behavior of a particular solution of \(y^{\prime}=1 / y\) as \(x \rightarrow \infty\) . To print an enlarged copy of the graph, go to MathGraphs.com. $$ \begin{array}{ll}{\text { (a) }(0,1)} & {\text { (b) }(1,1)}\end{array} $$

Short answer

The solution curves passing through the points (0,1) and (1,1) increase without bound as \( x \rightarrow \infty \) in the slope field of \( y'=1/y \).

Step-by-step solution

Sketch the solution passing through the point (0,1)

Recall that the slopes of the solution curves are given by the differential equation. In this case \( y'=1/y \), which implies that the slope of the solution curve at any point is the reciprocal of the y-coordinate of that point. Look at the slope field and find the point (0,1). Follow the slopes given by the field to sketch the solution curve passing through (0,1). Due to slopes, the curve should go upwards as we move away from the origin.

Sketch the solution passing through the point (1,1)

Follow the same process used in Step 1, but this time, start from the point (1,1). Again, due to the slopes, the solution curve should move upwards, similar to the first curve.

Conjecture about the behavior as \( x \rightarrow \infty \)

Looking at the sketched solutions, it's clear that, as \( x \rightarrow \infty \), the solutions appear to increase without bound. This suggests that for a particular solution of \( y'=1/y \), the function \( y \) will become infinitely large as \( x \rightarrow \infty \), i.e., \( y \rightarrow \infty \) as \( x \rightarrow \infty \).

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are central to understanding a myriad of physical, chemical, and biological processes. For example, the differential equation given in the exercise,
\(y' = \frac{1}{y}\),
is a first-order equation because it involves only the first derivative of the unknown function \(y\). This specific equation expresses that the rate of change of \(y\) at any point is the reciprocal of \(y\) itself. Understanding how to interpret such equations is fundamental to predicting the behavior of systems over time.

Initial Conditions

Initial conditions in the context of differential equations specify the state of the system at a particular point. They are crucial for determining the exact solution of a differential equation. With the initial condition, such as \((0, 1)\) or \((1, 1)\), we can find the particular solution that not only satisfies the differential equation but also passes through the given point. This transforms a general solution with potentially many curves into a single, specific path that the system will follow.

Graph Sketching

Graph sketching involves drawing a visual representation of solutions to differential equations. A slope field is a tool that helps in this process by providing a snapshot of the direction field of a differential equation. When sketching the graph of \(y' = \frac{1}{y}\), with initial conditions such as in this exercise, a slope field allows us to visualize how the solution should look. It acts as a guide, showing where the graph goes higher or lower, and helps us draw a curve that aligns with those directions. This technique turns abstract equations into understandable figures.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input either gets very large or very small. It gives us insight into the long-term trends of a system's behavior based on its differential equation. In the exercise, the conjecture about the behavior of the function as \(x \rightarrow \infty\) states that \(y\) is expected to increase without bound. This means that, over time, the solution to \(y' = \frac{1}{y}\) is predicted to grow indefinitely as we move along the x-axis to the right. This concept is important for predicting the ultimate fate of systems modeled by differential equations, such as whether they stabilize, loop, or shoot off towards infinity.