Question

In Exercises \(12-22\) construct a direction field and plot some integral curves in the indicated rectangular region. $$ y^{\prime}=\frac{x y^{2}}{y-1} ; \quad\\{-2 \leq x \leq 2,-1 \leq y \leq 4\\} $$

Short answer

Question: Construct a direction field and plot some integral curves in the indicated rectangular region defined by the given bounds on x and y for the first-order differential equation: $y' = \frac{xy^2}{y-1}$ for $-2 \leq x \leq 2$, $-1 \leq y \leq 4$.

Step-by-step solution

Determine the differential equation

In this problem, we have the given first-order differential equation: $$ y' = \frac{xy^2}{y-1} $$

Identify the rectangular region

The problem specifies the rectangular region with the following limits: $$ {-2 \leq x \leq 2, -1 \leq y \leq 4} $$

Create a grid of points in the specified rectangular region

Start by drawing a grid of points evenly spaced in the defined rectangular region. This will help us visualize and create the direction field for the differential equation. Make sure the grid points in both x and y directions are chosen in a way to satisfy the limits: $$ {-2 \leq x \leq 2, -1 \leq y \leq 4} $$

Calculate the slopes at each grid point

For each point in the grid, calculate the slope of the tangent to the integral curve at that point using the given first-order differential equation: $$ y' = \frac{xy^2}{y-1} $$ It is essential to note that this function is not defined when y = 1. Therefore, we will have a discontinuity in our direction field along the line y = 1.

Draw the direction field

Using the slopes calculated at each grid point, draw small line segments representing the slope of the integral curve at that point. This will create a visual representation of the direction field and the integral curves.

Plot Integral Curves

Choose several points in the rectangular region and use the direction field to draw smooth curves (integral curves) that approximate the solutions to the given differential equation that pass through those points. Each integral curve should follow the slope information provided by the direction field. The solution of the exercise is the direction field graph with some integral curves plotted through multiple points within the rectangular region.

Key concepts

Integral Curves

Integral curves are essential to understanding differential equations. They represent the solutions of a differential equation graphically. For a given differential equation expressed as \( y' = f(x, y) \), an integral curve can be seen as a path that follows the slope field generated by this differential equation.

The curve essentially "integrates" the equation by following the slope throughout a specified region. In our example, the differential equation is \( y' = \frac{xy^2}{y-1} \). The integral curves will represent the paths on which a solution to this equation moves within the defined space.

To create these curves:

• Pick starting points within the rectangular region.
• Follow the direction field, tracing a smooth curve that aligns with the slopes at each point.
• Remember that these curves can also show discontinuities, especially where the equation becomes undefined, such as along the line \( y = 1 \).

Observing integral curves helps to gain insights into the general behavior of solutions without solving the equation explicitly.

First-order Differential Equation

A first-order differential equation is one that involves a function and its first derivative. It is a key component in understanding physical phenomenon that change over time or space. The equation given in the exercise is \( y' = \frac{xy^2}{y-1} \), which is in the standard form \( y' = f(x, y) \).

These equations are called "first-order" because they involve derivatives that are first derivatives. Solving such equations means identifying a function \( y(x) \) that satisfies the equation. Often, these solutions are graphed as integral curves or found analytically.

Key steps to handle first-order differential equations include:

• Identifying the equation and its form.
• Determining any regions where the equation might be undefined (like \( y - 1 = 0 \) in this case).
• Using techniques such as separation of variables or integrating factors, if solving analytically, or plotting direction fields and integral curves for a visual understanding.

For our equation, the main focus is on the graphical solution using the direction field approach.

Rectangular Region

A rectangular region helps to define the space where you evaluate and draw the direction field and integral curves. It frames the problem, setting boundaries for the x and y variables.

In our exercise, the rectangular region is specified as \( -2 \leq x \leq 2 \) and \( -1 \leq y \leq 4 \). These bounds provide:

• The range of input values for \( x \).
• The range for where \( y \) values should be considered.
• A focus area where the solutions are plotted and studied.

The boundaries help you construct a grid that aids in calculating slopes for the direction field. This ensures the graphical solutions—the direction field and integral curves—are well-fitting and comprehensive for the given problem statement.

Direction Field Graph

A direction field graph is a visualization tool that helps understand differential equations. It consists of small line segments or arrows indicating the slope or direction in which the solution \( y \) is moving.

For our problem, the direction field is constructed for \( y' = \frac{xy^2}{y-1} \) over a given rectangular region. Each point in this field represents a specific \( (x, y) \) pair, and the line segment at that point depicts the slope given by the equation.

To build a direction field:

• Select a grid covering the x and y intervals of the rectangular region.
• Calculate the slope \( y' \) at each grid point.
• Draw small segments oriented according to these slope values.

This graphical depiction assists in drawing integral curves and understanding the solution's behavior by offering a visual map of possible solution trajectories.