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Chapter 2: Problem 15

In Problems, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may be semistable in the sense mentioned in Example 6.) $\$$ $$ \frac{d x}{d t}=\left(x^{2}-4\right)^{2} $$

Short Answer

Expert verified
The critical points \( x = 2 \) and \( x = -2 \) are semistable, indicating local stability but longer-term instability.

Step by step solution

01

Identify Critical Points

The critical points of a differential equation are found where \( \frac{dx}{dt} = 0 \). For the equation \( \frac{dx}{dt} = (x^2 - 4)^2 \), we set \( (x^2 - 4)^2 = 0 \). Solving for \( x \), we find the critical points are \( x = 2 \) and \( x = -2 \), because \( x^2 - 4 = 0 \) leads to \( x = \pm 2 \).
02

Determine Stability

To determine the stability of each critical point, analyze the sign of \( \frac{dx}{dt} \) around these points. It can be observed that the expression \((x^2 - 4)^2 \geq 0\) is always non-negative. Thus, both critical points, \( x = 2 \) and \( x = -2 \), could be stable if it does not change its stability nature when \( x \) approaches from any side.
03

Examine Nature of Stability

Since \( (x^2 - 4)^2 \) is always non-negative and zero only at \( x = \pm 2 \), these critical points are approached but never crossed, implying stability (or semistability). For \( x > 2 \) or \( x < -2 \), the system moves away from critical points, showing instability in these regions.
04

Plot Slope Field and Solutions

Using a computer graphing system or graphing calculator, input the differential equation \( \frac{dx}{dt} = (x^2 - 4)^2 \) to create a slope field. The slope field should visually represent the behavior around the critical points. Overlay solution curves to show trajectories beginning from various initial conditions. Note especially how close solutions behave around the critical points \( x = 2 \) and \( x = -2 \).

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Key Concepts

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

Critical Points
In the study of differential equations, critical points are locations where the rate of change of the variable is zero. For our differential equation \[ \frac{dx}{dt} = (x^2 - 4)^2 \],critical points occur where \[ (x^2 - 4)^2 = 0 \].Solving this, we find that the critical points are at \( x = 2 \) and \( x = -2 \). These points are significant because they indicate where the system is at rest, with no change occurring in the variable \( x \). Understanding these points helps us explore the overall behavior of the system governed by the differential equation.
Slope Field
A slope field is a graphical representation that provides a visual insight into the behavior of a differential equation. Essentially, it consists of small line segments or arrows at various points on the plane, representing the slope of the solution at that point. For the equation \( \frac{dx}{dt} = (x^2 - 4)^2 \),the slope field would feature flat lines (slope of 0) at the critical points \( x = 2 \) and \( x = -2 \), since this is where the system's rate of change is zero.
  • Visualizing slope fields can help you see how solutions behave between critical points and beyond.
  • Slope fields sometimes also reveal whether a critical point is stable, unstable, or semistable by showing the direction of the flow around these points.
Plotting slope fields is a powerful technique to understand differential equations without analytically solving them.
Stability Analysis
Stability in critical points refers to whether solutions that start near the critical point tend to return to it, diverge away, or neither. This behavior is crucial in understanding the nature of the solution to a differential equation. For our equation\( \frac{dx}{dt} = (x^2 - 4)^2 \),we observe that the function is non-negative and zero only at the critical points \( x = 2 \) and \( x = -2 \). Solution trajectories around these critical points indicate:
  • Stability: Near the critical points, the slope is zero, suggesting these points are equilibrium states where the system rests.
  • Instability: For values \( x > 2 \) or \( x < -2 \), the slope field reveals that solutions divert away, indicating instability in these regions.
This analysis shows how critical points function as attractors or repellers in the system.
Solution Curves
Solution curves are paths that different initial conditions will follow over time when traced out on the slope field of a differential equation. They help illustrate how the system evolves from one state to another. Using a computer graphing tool, we can plot \( \frac{dx}{dt} = (x^2 - 4)^2 \)to observe the solution curves:
  • Near \( x = 2 \) and \( x = -2 \): The solution curves approach these points but do not cross them, reinforcing their role as stable points.
  • Far from \( x = 2 \) and \( x = -2 \): Solution curves demonstrate divergence, showcasing instability beyond these critical regions.
Examining solution curves provides a comprehensive view of the dynamics within the system, highlighting how proximity to critical points affects the system's behavior.

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Most popular questions from this chapter

A motorboat starts from rest (initial velocity \(v(0)=v_{0}=\) 0). Its motor provides a constant acceleration of \(4 \mathrm{ft} / \mathrm{s}^{2}\), but water resistance causes a deceleration of \(v^{2} / 400 \mathrm{ft} / \mathrm{s}^{2}\) Find \(v\) when \(t=10 \mathrm{~s}\), and also find the limiting velocity as \(t \rightarrow+\infty\) (that is, the maximum possible speed of the boat).

As in Problem 25 of Section \(2.4\), you bail out of a helicopter and immediately open your parachute, so your downward velocity satisfies the initial value problem $$ \frac{d v}{d t}=32-1.6 v, \quad v(0)=0 $$ (with \(t\) in seconds and \(v\) in \(\mathrm{ft} / \mathrm{s}\) ). Use the improved Euler method with a programmable calculator or computer to approximate the solution for \(0 \leqq t \leqq 2\), first with step size \(h=0.01\) and then with \(h=0.005\), rounding off approximate \(v\) -values to three decimal places. What percentage of the limiting velocity \(20 \mathrm{ft} / \mathrm{s}\) has been attained after 1 second? After 2 seconds?

In Problems, use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may be semistable in the sense mentioned in Example 6.) $\$$ $$ \frac{d x}{d t}=x\left(x^{2}-4\right) $$

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval \([0,0.5]\) with step size \(h=0.1 .\) Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points \(x=0.1,0.2,0.3\), \(0.4,0.5\) $$ y^{\prime}=\frac{1}{4}\left(1+y^{2}\right), y(0)=1 ; y(x)=\tan \frac{1}{4}(x+\pi) $$

A population \(P(t)\) of small rodents has birth rate \(\beta=\) (0.001) \(P\) (births per month per rodent) and constant death rate \(\delta\). If \(P(0)=100\) and \(P^{\prime}(0)=8\), how long (in months) will it take this population to double to 200 rodents? (Suggestion: First find the value of \(\delta\).)
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