Scroll back to the top

Virtual Math Learning Center Logo

Section 2.5 – Autonomous Equations and Population Dynamics

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. 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. \(\begin{cases}  y_0<-2 & \Rightarrow & y\to -\infty \\ y_0=-2 & \Rightarrow & y=-2 \\ -2< y_0<0 & \Rightarrow & y\to 0 \\ y_0=0 & \Rightarrow & y= 0 \\ 0< y_0<2 & \Rightarrow & y\to 0 \\y_0=2 & \Rightarrow & y=2 \\ y_0>2 & \Rightarrow & y\to \infty  \end{cases}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V22


  2. 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.

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V23