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Chapter 2: Problem 2

In Problems first solve the equation \(f(x)=0\) to find the critical points of the given autonomous differential equation \(d x / d t=f(x) .\) Then analyze the sign of \(f(x)\) to determine whether each crifical point is stable or unstable, and construct the corresponding phase diagrant for the differential equarion. Next, solve the differential equarion explicitly for \(x(t)\) in terms of t. Finally use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point. $$ \frac{d x}{d t}=3-x $$

Short Answer

Expert verified
The critical point is \(x = 3\), and it is stable. The solution \(x(t) = 3 - Ce^{-t}\) shows convergence to \(x = 3\).

Step by step solution

01

Find Critical Points

To find critical points, solve for \(x\) in the equation \(f(x) = 3 - x = 0\), setting \(f(x)\) to zero. This gives \(x = 3\) as the sole critical point.
02

Analyze the Sign of f(x)

The sign of \(f(x) = 3 - x\) changes around the critical point. For \(x < 3\), \(f(x) = 3 - x > 0\), meaning \(dx/dt > 0\) and solutions move right (toward \(x = 3\)). For \(x > 3\), \(f(x) = 3 - x < 0\), meaning \(dx/dt < 0\) and solutions move left (toward \(x = 3\)). Thus, the critical point \(x = 3\) is stable.
03

Construct the Phase Diagram

On a phase line, mark the critical point \(x = 3\). Draw arrows pointing toward \(x = 3\) from both sides to signify stability and convergence toward the point.
04

Solve the Differential Equation

Separate the variables and integrate both sides: \(\int \frac{1}{3-x} \, dx = \int \, dt\). This gives \(-\ln|3-x| = t + C\). Solving for \(x\), we get \(x(t) = 3 - Ce^{-t}\).
05

Sketch Solution Curves and Verify Stability

Plot the solutions \(x(t) = 3 - Ce^{-t}\) for different values of \(C\). The solutions approach \(x = 3\) as \(t \to \infty\), demonstrating stability as they all converge to the critical point.

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Key Concepts

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

Critical Points in Autonomous Differential Equations
In an autonomous differential equation like \( \frac{dx}{dt} = f(x) \), finding critical points is a crucial first step. Critical points, or equilibrium points, occur where the rate of change is zero, meaning \( f(x) = 0 \). For the equation \( \frac{dx}{dt} = 3 - x \), setting \( f(x) = 0 \) reveals one critical point at \( x = 3 \). These points are vital as they indicate where a system's state doesn't change over time. If a system reaches a critical point, it remains there unless disrupted. Recognizing these points allows us to predict system behavior over time.
Stability Analysis of Critical Points
After identifying critical points, we determine their stability. A stable critical point attracts nearby solutions over time, while an unstable one repels them. For \( f(x) = 3 - x \), analyzing the sign around the critical point reveals its stability. - When \( x < 3 \), \( f(x) = 3 - x > 0 \), indicating that \( \frac{dx}{dt} > 0 \), and solutions move right.- When \( x > 3 \), \( f(x) = 3 - x < 0 \), indicating that \( \frac{dx}{dt} < 0 \), and solutions move left.Both cases show solutions converging toward \( x = 3 \), marking it as a stable point. Stability analysis helps understand whether a system will naturally settle at a critical point or diverge away.
Phase Diagram Construction
A phase diagram visually represents system dynamics through arrows on a phase line. This tool helps conceptualize stability and motion direction in a system. For \( \frac{dx}{dt} = 3 - x \), we create a phase diagram by:- Plotting the critical point \( x = 3 \) on a phase line.- Drawing arrows toward \( x = 3 \) from both sides, illustrating that solutions converge to this stable point.Phase diagrams are simple, yet powerful illustrations that make the tendency of a system to evolve toward or away from equilibrium very clear. By understanding these visuals, learners can vividly "see" system behavior.
Solution Curves and Verification
Finally, solving the differential equation explicitly demonstrates the behavior of solution curves over time. For \( \frac{dx}{dt} = 3 - x \), we separate variables and integrate, yielding \( x(t) = 3 - Ce^{-t} \). Solution curves plot \( x(t) \) relative to time \( t \): - For various constants \( C \), curves start from different points but all asymptotically approach \( x = 3 \) as time increases.This convergence visually and analytically confirms the stability of the critical point. Understanding solution curves complements stability analysis and phase diagrams, providing a comprehensive view of how initial conditions evolve over time.

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Most popular questions from this chapter

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.) $\$$ $$ \frac{d x}{d t}=\left(x^{2}-4\right)^{2} $$

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 \(\delta\).)

(a) Show that if a projectile is launched straight upward from the surface of the earth with initial velocity \(v_{0}\) less than escape velocity \(\sqrt{2 G M / R}\), then the maximum distance from the center of the earth attained by the projectile is $$ r_{\max }=\frac{2 G M R}{2 G M-R v_{0}^{2}} $$ where \(M\) and \(R\) are the mass and radius of the earth, respectively. (b) With what initial velocity \(v_{0}\) must such a projectile be launched to yield a maximum altitude of 100 kilometers above the surface of the earth? (c) Find the maximum distance from the center of the earth, expressed in terms of earth radii, attained by a projectile launched from the surface of the earth with \(90 \%\) of escape velocity.

Separate variables and use partial fractions to solve the initial value problems in Problems \(1-8 .\) Use either the exact solution or a computer- generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution. $$ \frac{d x}{d t}=7 x(x-13), x(0)=17 $$

A programmable calculator or a computer will be useful for Problems 11 through \(16 .\) In each problem find the exact solution of the given initial value problem. Then apply the RungeKutta method twice to approximate (to five decimal places) this solution on the given interval, first with step size \(h=0.2\), then with step size \(h=0.1 .\) Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for \(x\) an integral multiple of 0.2. Throughout, primes denote derivatives with respect to \(x\). \(y^{\prime}=\frac{1}{2}(y-1)^{2}, y(0)=2 ; 0 \leqq x \leqq 1\)
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