The most classic of all second-order differential equations is the harmonic oscillator: $$m\frac{d^2y}{dt^2}+b\frac{dy}{dt}+ky=0.$$ This is a second-order, homogeneous, linear, ordinary differential equation with constant coefficients. For this assignment, we assume that, $m>0$, $b\ge0$ and $k>0$. This equation models a spring-mass system, where $m$, $b$ and $k$ represent, respectively, the mass, the damping coefficient and the spring stiffness. The dependent variable $y$ describes the displacement of the mass, that is, how much the spring is stretched or compressed.

We adopt the point of view that a second-order differential equation can be reduced to a system of two first order differential equations. In our case the system is: $$\frac{dy}{dt}=v,\quad\ \frac{dv}{dt}=-\frac{k}{m}y-\frac{b}{m}v .$$

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

Notice that the parameters $m$, $b$ and $k$ are not properly set. The first thing you should do is to choose appropriate values. Examine the solutions for at least three different sets of parameters. The first equation you should study is the harmonic oscillator with no damping, that is, with $b=0$. Next try to find parameters for differential equations whose solutions have distinctly different qualitative behavior. Report your results by answering each question below in the answer box that follows the question. Be sure to include the parameters and the initial points of your curves in your discussion.

And remember to click the "Submit answers" button when you are done!

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

Redraw using the new parameters and 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.