Chapter 1: Problem 21
In Exercises \(12-22\) construct a direction field and plot some integral curves in the indicated rectangular region. $$ y^{\prime}=\frac{x\left(y^{2}-1\right)}{y} ; \quad\\{-1 \leq x \leq 1,-2 \leq y \leq 2\\} $$
Short Answer
Expert verified
Based on the given first-order differential equation and rectangular region, explain the process of constructing a direction field and plotting integral curves.
Step by step solution
01
Analyze the given differential equation
We are given the first-order differential equation:
$$
y^{\prime}=\frac{x\left(y^{2}-1\right)}{y}
$$
Define the function \(f(x, y)\) representing the slopes at each point:
$$
f(x, y) = \frac{x\left(y^{2}-1\right)}{y}
$$
02
Create a rectangular grid
By the given rectangular region \(-1 \leq x \leq 1,\ -2 \leq y \leq 2\), we create a grid with equally spaced points in the region. For example, you can choose a grid with 11 equally spaced points along the x-axis and 21 equally spaced points along the y-axis.
03
Calculate the slopes at each grid point
For each point \((x, y)\) in the grid, find the slope of the tangent at this point by evaluating the function \(f(x, y)\):
$$
m = f(x, y) = \frac{x\left(y^{2}-1\right)}{y}
$$
The slope \(m\) will give the direction of the tangent line at the point \((x, y)\).
04
Draw the tangent lines at each grid point
Using the calculated slopes \(m\), draw short tangent lines at each point \((x, y)\) in the grid. The lines should be drawn with a small length to avoid overlapping with other tangent lines. These short tangent lines represent the direction field.
05
Plot integral curves
Now that the direction field is ready, we can plot integral curves. To do this, pick a point in the rectangular grid as the initial condition \((x_0, y_0)\). Then, follow the tangent lines in the direction field to trace the integral curve that passes through the initial condition.
Repeat this process for various initial conditions to plot a variety of integral curves.
Once the integral curves have been plotted, the direction field and the integral curves will give a visual representation of the behavior of the solutions to the given differential equation in the-defined rectangular region.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Direction Field
A Direction Field provides a visual representation of a differential equation. It consists of small line segments or arrows drawn at grid points in a plane. Each segment shows the slope of the solution to the differential equation at that point. By piecing these segments together, we can visualize how solutions behave.
To construct a direction field:
- Start by creating a grid in the specified region of the equation.
- At each grid point, compute the slope using the differential equation.
- Draw a small line segment with this slope, representing the tangent to an integral curve.
Integral Curves
Integral Curves are paths that follow the direction field. They represent solutions to differential equations and show how solutions evolve over time or along the x-axis. When you draw an integral curve:
- Select an initial point on the direction field.
- Trace along the tangent slopes, following the arrows or lines.
First-order Differential Equation
A First-order Differential Equation involves derivatives of the first degree. For such equations, the derivative usually depends on both the x and y variables. In the given example, the equation is displayed as:\[y^{\prime} = \frac{x\left(y^{2}-1\right)}{y}\]Understanding first-order differential equations involves determining how changes in x affect changes in y, facilitated by finding slopes using the equation.Key points include:
- A solution to a first-order differential equation is a function or curve satisfying the equation.
- These equations can model various real-world processes like growth rates or mechanical systems.
- Direction fields and integral curves are vital visual tools that provide insight into their solutions.
Slope Field
Slope Fields are synonymous with direction fields. They provide a graphical representation of many possible slopes at points
across a plane, describing the solutions to a differential equation. In simpler terms, it visualizes the solution
landscape of the differential equation. Each slope corresponds to:
across a plane, describing the solutions to a differential equation. In simpler terms, it visualizes the solution
landscape of the differential equation. Each slope corresponds to:
- The solution's rate of change at given points.
- Helping us predict solution curves by showing the likely path or trajectory in the plane.
- Calculate the slope using given x and y values in the equation.
- Draw small line segments representing these slopes on the grid.