Section 2.5 – Autonomous Equations and Population Dynamics
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. Given the differential equation \[y'=y^3-4y\]
- Find the equilibrium solutions.
- Graph the phase line. Classify each equilibrium solution as either stable, unstable, semistable.
- Graph some solutions.
- 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.
- Equilibrium Solutions: \(y=0,\) \(y=2,\) \(y=-2\)
- Stable: \(y=0,\) Unstable: \(y=2,\) \(y=-2.\) See the video for the graph of the phase line
- See the video
- \(\begin{cases} y_0<-2 & \Rightarrow & y\to -\infty \\ y_0=-2 & \Rightarrow & y=-2 \\ -2< y_0<0 & \Rightarrow & y\to 0 \\ y_0=0 & \Rightarrow & y= 0 \\ 0< y_0<2 & \Rightarrow & y\to 0 \\y_0=2 & \Rightarrow & y=2 \\ y_0>2 & \Rightarrow & y\to \infty \end{cases}\)
2. Given the differential equation \[y'(t)=y^3-2y^2+y\]
- Find the equilibrium solutions.
- Graph the phase line. Classify each equilibrium solution as either stable, unstable, or semistable.
- Sketch the graph of some solutions.
- 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.
- Do any solutions 'blow up in finite time,' namely, do they admit a vertical asymptote?
- Equilibrium Solutions: \(y=0,\) \(y=1,\)
- Semistable: \(y=1,\) Unstable: \(y=0,\) See the video for the graph of the phase line
- See the video
- \(\begin{cases} y_0<0 & \Rightarrow & y\to -\infty \\ y_0=0 & \Rightarrow & y=0 \\ 0< y_0<1 & \Rightarrow & y\to 1 \\ y_0=1 & \Rightarrow & y= 1 \\ y_0>1 & \Rightarrow & y\to \infty\end{cases}\)
- Yes, \(y(t)\) blows up in finite time. See the video for an explanation.