About Differential Equations

Abstract

The subject of differential equations is large, diverse, powerful, useful, and full of surprises. Differential equations can be studied on their own—just because they are intrinsically interesting. Or, they may be studied by a physicist, engineer, biologist, economist, physician, or political scientist because they can model (quantitatively explain) many physical or abstract systems. Just what is a differential equation? A differential equation having y as the dependent variable (unknown function) and x as the independent variable has the form

$$F(x,y,\frac{{dy}}{{dx}},...,\frac{{{d^n}y}}{{d{x^n}}}) = 0$$

for some positive integer n. (If n is 0, the equation is an algebraic or transcendental equation, rather than a differential equation.) Here is the same idea in words: