[iano]

In short: Learn constant coefficient 2nd order differential equations, and you can design a whole slew of mechanical, electrical, and hydraulic contraptions.

A real application of this: before computers, mechanical systems control engineers would write the equations of their mechanical systems, write their controllers, and then implement their controllers around the electrical circuit that duplicated the behavior of their mechanical system. Sort of an analog computer way of simulating a system.

[timclark]

Or as in my experience as an EE, learn how to use Laplace transforms, find a table for the 2nd order differential equations, use Laplace transforms, forget how to solve differential equations.