Linear Algebra, 5th Edition

2.7* Homogeneous Linear Differential Equations with Constant Coefficients

As an introduction to this section, consider the following physical problem. A weight of mass m is attached to a vertically suspended spring that is allowed to stretch until the forces acting on the weight are in equilibrium. Suppose that the weight is now motionless and impose an xy-coordinate system with the weight at the origin and the spring lying on the positive y-axis (see Figure 2.7).

Figure 2.7

Suppose that at a certain time, say $t=0$, the weight is lowered a distance s along the y-axis and released. The spring then begins to oscillate.

We describe the motion ...