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Why are differential equations called differential equations?

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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).

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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.