Question

In Exercises \(12-22\) construct a direction field and plot some integral curves in the indicated rectangular region. $$ y^{\prime}=\frac{x}{y\left(y^{2}-1\right)}: \quad\\{-2 \leq x \leq 2,-2 \leq y \leq 2\\} $$

Short answer

Question: Briefly describe the main steps involved in constructing a direction field and plotting some integral curves for the given differential equation: \(y^{\prime} = \frac{x}{y(y^2 - 1)}\) Answer: The main steps include 1) setting up the differential equation, 2) calculating the slope for each point in the grid, 3) drawing the slope field (direction field), and 4) plotting some integral curves within the specified rectangular region.

Step-by-step solution

Set up the differential equation

Given the differential equation: $$ y^{\prime}=\frac{x}{y\left(y^{2}-1\right)} $$ We need to plot the direction field and integral curves in the given rectangular region \({-2 \leq x \leq 2, -2 \leq y \leq 2}\).

Calculate the slope for each point in the grid

We will calculate the slope of the tangent line to the curve at various points in the rectangular region specified. The slope of the tangent line is given by the value of the function \(y^{\prime}\) at that point. For example, if we choose the point \((x, y) = (0, 1)\), we can calculate the slope as follows: $$ y^{\prime}(0,1) = \frac{0}{1\left(1^{2} - 1\right)} = \text{undefined because the denominator is 0} $$ We will continue to calculate the slope of the tangent line at other points in the grid systematically (say for example, in intervals of 0.5)

Draw the slope field (direction field)

Now that we have calculated the slopes at various points in our grid, we can plot the direction field. Draw small lines representing the slope of the tangent line at each point we calculated the slope in step 2. This will give us a sense of how the solution curves should behave in the given rectangular region.

Plot some integral curves

The final step is to plot some integral curves, which are the curves that satisfy the given differential equation in the rectangular region \({ -2 \leq x \leq 2, -2 \leq y \leq 2 }\). For this, we will start at an initial point in the grid, say \((x_0, y_0)\) and use the direction field from step 3 as a guide to draw the curve going through that initial point and following the direction of the slope lines. We will then repeat this process for various initial points in the grid to give several integral curves.

Key concepts

Differential Equation

A differential equation is a mathematical equation that involves functions and their derivatives. In the context of direction fields, a differential equation usually describes how a quantity changes over time or space. The given problem provides the differential equation \( y' = \frac{x}{y(y^2 - 1)} \). Here, \( y' \) represents the derivative of \( y \) with respect to \( x \). This equation can be complex as its behavior changes depending on the values of \( x \) and \( y \).

Understanding differential equations helps in modeling real-world scenarios. For instance:

• It can describe how populations grow in ecology.
• It can model physical systems like motion in physics.

Solving these equations might require different techniques, like plotting direction fields, to visualize possible solutions.

Slope Field

A slope field, also known as a direction field, is a visual representation of a differential equation. It consists of small line segments or arrows that show the slope of the solution curve at given points in the plane. In this problem, we need to create a slope field for the differential equation \( y' = \frac{x}{y(y^2 - 1)} \). This involves:

• Choosing points in the given rectangular region \(-2 \leq x \leq 2, -2 \leq y \leq 2\).
• Calculating the slope \( y' \) at each point and drawing a short line to represent it.

These lines suggest how a curve should flow through a particular area. In areas where the slope is undefined, you might see breaks in the field, indicating critical points where the behavior changes.

Integral Curves

Integral curves are the solution curves of a differential equation. They follow the direction of the slope field, representing possible paths that a system could follow over time. In our problem, after drawing the slope field, integral curves are sketched starting from initial points within the region \(-2 \leq x \leq 2, -2 \leq y \leq 2\).

This involves:

• Selecting starting points from the grid.
• Using the direction of the slopes to trace a smooth curve.

These curves help visualize how solutions behave, showing whether they converge, diverge, or follow other patterns based on the differential equation. Integral curves provide insights into how the system changes and can be used to predict future behavior.

Rectangular Region

A rectangular region defines the area where the slope field and integral curves are plotted. In our problem, this region is \(-2 \leq x \leq 2, -2 \leq y \leq 2\). This specific boundary helps focus the analysis and visualization.

When constructing the direction field:

• Choose points within the rectangle to calculate the slope.
• Ensure the curves begin and remain within this region.

The rectangular boundary is crucial for setting up a controlled environment to study the behavior of the differential equation. It allows us to limit calculations to a manageable space, providing a clear picture of how the integral curves behave in that specified area.