Consider the differential equation $$m\frac{d^2y}{dt^2}+b\frac{dy}{dt}+ky=0.$$ Assume $m=5$, $b=4$ and $k=4$; that is, we'll focus on the equation: $$5\frac{d^2y}{dt^2}+4\frac{dy}{dt}+4 y=0,$$ a second-order, homogeneous, linear, ordinary differential equation with constant coefficients.

Below is a phase-plane plot of the vector field $\big(y'(t),v'(t)\big)$ in the $y,v$-plane, where $v$ is defined by the equation $$y'(t) = v(t)$$ and therefore, according to the formula for the differential equation, $$v'(t) = y''(t) = -\frac{4}{5}y-\frac{4}{5}v.$$ If $y(t)$ is a solution to the differential equation then the curve $\big(y(t),y'(t)\big) = \big(y(t),v(t)\big)$ is tangent to the vector field, so one can find solutions by looking for curves that are tangent to the vector field.

Examine solutions using the phase plane below. Click on the phase plane to graph solutions $\big(y(t),v(t)\big)$; the clicked point is the initial point $\big(y(0),v(0)\big)$. You may change the range of $t$ that you wish to plot, then push the "Redraw ..." button to implement the change.

Then answer the questions below by writing your answers in the boxes below the questions. Remember to click the "Submit answers" button when you are done!

Set new bounds for $t$: min = , max = with min $\leq$ 0 $\leq$ max.

Redraw using the new bounds Clear the solution curves

Questions:

Are solutions to this system periodic? Explain your answer in terms of the phase plane plot.

If there are periodic solutions, choose a particular initial condition and estimate the period of the corresponding solution to two decimal places. Explain how you found this value.

Do all solutions have the same period? How can you decide that using solutions plotted in the phase plane?

Write a paragraph summarizing the observations you made. Your summary should include a description of the behavior of the solutions for this system for different initial conditions, and compare similarities and differences between distinct solutions.

Adapted from Differential Equations, 4th Ed., Blanchard, Devaney, Hall, 2012.