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Section 8.2 - Equilibria and Their Stability

Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. Find and classify the equilibria for the equation.\[\dfrac{dy}{dt}=y^3-4y\]

\(y_1=0,\) stable;  \(y_2=2,\) unstable;  \(y_3=-2,\) unstable

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2. Find and classify the equilibria for the equation.\[\dfrac{dy}{dt}=y^3-2y^2+y\]

\(y_1=0,\) unstable;  \(y_2=1,\) semistable

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3. Find and classify the equilibria for the equation.\[\dfrac{dy}{dt}=y^3\]

\(y_1=0,\) unstable

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4. Given the differential equation \[y'=y^3-4y\]

  1. Find the equilibrium solutions.
  2. Graph the phase line. Classify each equilibrium solution as either stable, unstable, semistable.
  3. Graph some solutions.
  4. If \(y(t)\) is the solution of the equation satisfying the initial condition \(y(0)=y_0\), where \(-\infty< y_0<\infty\), find the limit of \(y(t)\) when \(t\) increases.

  1. Equilibrium Solutions: \(y=0,\) \(y=2,\) \(y=-2\)
  2. Stable: \(y=0,\) Unstable: \(y=2,\) \(y=-2.\) See the video for the graph of the phase line
  3. See the video
  4. \(\begin{cases}  y_0<-2 & \Rightarrow & y\to -\infty \\ y_0=-2 & \Rightarrow & y=-2 \\ -2< y_0<0 & \Rightarrow & y\to 0 \\ y_0=0 & \Rightarrow & y= 0 \\ 0< y_0<2 & \Rightarrow & y\to 0 \\y_0=2 & \Rightarrow & y=2 \\ y_0>2 & \Rightarrow & y\to \infty  \end{cases}\)

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5. Given the differential equation \[y'(t)=y^3-2y^2+y\]

  1. Find the equilibrium solutions.
  2. Graph the phase line. Classify each equilibrium solution as either stable, unstable, or semistable.
  3. Sketch the graph of some solutions. 
  4. If \(y(t)\) is the solution of the equation satisfying the initial condition \(y(0)=y_0\), where \(-\infty< y_0<\infty\), determine the behavior of \(y(t)\) as \(t\) increases.
  5. Do any solutions 'blow up in finite time,' namely, do they admit a vertical asymptote?

  1. Equilibrium Solutions: \(y=0,\) \(y=1,\) 
  2. Semistable: \(y=1,\) Unstable: \(y=0,\) See the video for the graph of the phase line
  3. See the video
  4. \(\begin{cases}  y_0<0 & \Rightarrow & y\to -\infty \\ y_0=0 & \Rightarrow & y=0 \\ 0< y_0<1 & \Rightarrow & y\to 1 \\ y_0=1 & \Rightarrow & y= 1 \\ y_0>1 & \Rightarrow & y\to \infty\end{cases}\)
  5. Yes, \(y(t)\) blows up in finite time. See the video for an explanation.

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