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Chapter 7: Problem 16

Plot the direction field of the differential equation. \(\frac{d y}{d x}=-4 x y\)

Short Answer

Expert verified
Plot the direction field by calculating slopes \(-4xy\) at grid points and drawing line segments accordingly.

Step by step solution

01

Understanding the Differential Equation

The given differential equation is \( \frac{dy}{dx} = -4xy \). This is a first-order differential equation, where the rate of change of \( y \) with respect to \( x \) is equal to \(-4xy\). To plot a direction field, we need to assess how the slope of the solution \( y \) changes for different values of \( x \) and \( y \).
02

Setting Up the Slope Formula

In a direction field, the slope or direction is represented by small line segments for various points \((x, y)\). The slope of these line segments is given by \( \frac{dy}{dx} \). For this equation, the slope is \( -4xy \).
03

Choosing Values of \(x\) and \(y\)

To plot the direction field, choose a grid of points over which you will calculate and draw the slope. Let's choose integer values of \( x \) and \( y \) in the range from -2 to 2, inclusive.
04

Calculating Slopes for Each Grid Point

For each chosen point \((x, y)\), substitute these values into the equation \( -4xy \) to find the slope. For example, for \((x, y) = (1, 1)\), the slope is \(-4(1)(1) = -4\). For \((x, y) = (-1, 1)\), the slope is \(-4(-1)(1) = 4\). Continue this process for all grid points.
05

Plotting the Direction Field

Using a coordinate plane, draw small line segments centered at each grid point using the slopes calculated. A positive slope (e.g., 4) will have an upward direction, while a negative slope (e.g., -4) will have a downward direction. Repeat this for all grid points to visualize the general direction of potential solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-order Differential Equation
A first-order differential equation is an equation involving the derivative of a function and the function itself. In the equation \( \frac{dy}{dx} = -4xy \), the term \( \frac{dy}{dx} \) represents the rate of change of \( y \) with respect to \( x \). This particular equation is linear in terms of its highest derivative, which is the first derivative here. The term \( -4xy \) shows how the change in \( y \) is influenced by both \( x \) and \( y \) values. First-order equations often appear in natural phenomena and are fundamental in understanding dynamic systems. They help us determine how a small change in one variable affects another.
Slope Formula
In the context of differential equations, the slope formula specifies the steepness or direction of a line at any given point on a graph. For the equation \( \frac{dy}{dx} = -4xy \), the slope is calculated as \( -4xy \). This slope formula is crucial when plotting a direction field, as it gives the direction of the flow of possible solutions.

Key points about the slope formula:
  • It provides the rate at which \( y \) changes with respect to \( x \).
  • Different \( x, y \) pairs will yield different slopes.
  • Understanding this can predict the behavior of the system being modeled.
Using this information, small line segments can be drawn on a graph, showing the direction and inclination of potential solution curves.
Grid Points
Grid points are crucial for creating a direction field. They represent specific locations on a coordinate plane where the slope is calculated and displayed. When choosing grid points for a differential equation, it's essential to cover a range of \( x \) and \( y \) values to get a comprehensive view of the possible solutions' behavior. In our exercise, integer values of \( x \) and \( y \) from -2 to 2 are used.

Key aspects of using grid points include:
  • Providing a structured framework to evaluate the slope at each point.
  • Allowing visualization of the direction field, which overall helps in understanding the possible trajectories of solutions.
  • Covering a symmetric range so that behavior around the origin can be assessed accurately.
These grid points form the basis for plotting individual directional segments on the graph.
Differential Equation Solution
While a direction field provides a graphical view of the differential equation's behavior, solving a differential equation involves finding a specific function \( y(x) \) that satisfies the equation. For the equation \( \frac{dy}{dx} = -4xy \), a solution \( y(x) \) would be a function whose derivative matches the given expression when substituted back into the differential equation. There are several methods for finding solutions:
  • Analytical methods, which involve integrating the adjusted equation to find a particular solution.
  • Numerical methods, like Euler's method, when analytical integration is complex or impossible.
The solution can help in predicting future behavior or patterns in the modeled scenario. Thus, gaining insight from the direction field helps guide the solution of the equation by showing potential patterns or steady states.

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Most popular questions from this chapter

Use l'H么pital's Rule to find the limit. $$ \lim _{x \rightarrow \infty} \frac{1}{x\left(\pi / 2-\tan ^{-1} x\right)} $$

A differential equation of the form \(d y / d x=f(y / x)\) is called homogeneous. a. Let \(v=y / x\). Show that $$ \frac{d y}{d x}=x \frac{d v}{d x}+v $$ b. Use (15) to show that a homogeneous differential equation \(d y / d x=f(y / x)\) can be rewritten in the form $$ \frac{1}{f(v)-v} d v=\frac{1}{x} d x $$

Prove that $$ \tan ^{-1} \frac{x}{\sqrt{1-x^{2}}}=\sin ^{-1} x \text { for }-1

Prove that \(\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\) provided that the value of the expression on the lefthand side lies in \([-\pi / 2, \pi / 2]\).

Use l'H么pital's Rule to find the limit. $$ \lim _{x \rightarrow \infty} \frac{\log _{4} x}{x} $$
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