Giant brains; or, Machines that think

by Edmund Callis Berkeley

Or we can say, and it is just the same thing in other words:

(speed) equals the derivative of (distance) with respect to (time)

Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.

SOLVING PHYSICAL PROBLEMS

Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the inverse or opposite of multiplication: division.

In the same way, we can change the equation about speed into other forms, if we want to find distance or time instead of speed. If we do this, we run into a new idea, the inverse or opposite of the derivative, called integral. The two new equations are:

(distance) equals the integral of (speed) with respect to (time)

(time) equals the integral of [one divided by (speed)] with respect to (distance)

These equations may also be called differential equations.

An integral is the result of a process called integrating. To integrate speed and get distance is the result of three steps: (1) breaking up an interval of time into a large number of small bits, (2) adding up all the small distances that we get by taking each bit of time and multiplying by the speed which applied in that bit of time, and (3) letting the bits of time get smaller and smaller, and letting the number of them get larger and larger, indefinitely.