Why are differential equations called differential equations? Why are differential equations called differential equations?
 A: Because they are equations (with the variable being a function, not a number) that involve a function and its derivatives (the functions obtained by differentiating it).
A: Why is a differential equation called a differential equation?
Here is an answer in a conceptual form.
You are used to seeing a curve described directly as a function y=f(x).
But every point on a curve also has a linear slope. If you 
know the slope at every point on a curve (and a starting point), then you can 
reproduce the curve. Think of all the linear slopes as forming an
envelope. Now if you know the slope of a curve at every point, then you
can form an equation relating a delta x at a point to a delta y at the
point, delta y = slope(x) * delta x.  The delta x and delta y 
are differentials, and thus you have an equation relating a 
differential in x at a point to a differential in y at a point - 
that is, a differential equation! Usually this equation is 
rearranged as delta y/delta x = slope(x), or in the limit, 
dy/dx = slope(x), or dy/dx = f(x), where f(x) is the slope.
Now solving a differential equation
means finding the original curve that has the specified slope
at each point. This is done by integration. In a finite approximation,
with a starting point of zero, this would be, for example,
f(x)=delta x * slope(x1) + delta x * slope(x2) + delta x * slope(x3)...
and letting delta x shrink toward zero gives a better approximation. 
But this formula is simply integration. 
Integration finds the area under a curve, but the area
under the curve is inherently the solution to a 
differential equation. Integrating inherently solves some 
differential equation.
