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:

Examine solutions using the phase plane below. Click on the phase plane to
graph solutions

Notice that the parameters

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

Set new parameters:

Set new bounds for

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.

You can earn partial credit on this problem.