[femto]

In my EE work, I tend to use formal analogies in my thought processes, sometimes unconsciously. For example, water and electricity behave in same way [1]. You don't design the whole circuit in the water domain, but you might temporarily flip over to water to understand a particular aspect, then flip back to electrical. Typically I won't write anything down in the alternative domain, using it just enough to clarify my thinking in the electrical domain, then recording things in electrical terms. There's all sorts of other domains too: acoustics, mechanics, pneumatics, ... Pick the one you are most comfortable with in the situation.

[1] http://en.wikipedia.org/wiki/Hydraulic_analogy

[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.