A block is connected to two fixed walls by a spring on one side and a damper on the other. The equation of motion is

,

where and are the spring stiffness and dampening coefficients, is the mass of the block, is the displacement of the mass, and is the time. This example deals with the underdamped case only.

This example comes from [1], Section 4.1, Modeling with First Order Equations.

Reference

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.