IV.

Differential Equations

Calculus leads directly to the branch of mathematics called differential equations, which is extremely useful in engineering and in the physical sciences. An ordinary differential equation is an equation involving an independent variable, a dependent variable (one or both of these two may be missing), and one or more derivatives (at least one derivative must be present). Many physical laws or statements are initially expressed as differential equations. For example, the law that the acceleration of gravity is a constant g can be expressed mathematically by the differential equation d²x/dt² = g; the principle that the rate of disintegration of radium is proportional to the amount present is expressed as dR/dt = -kR. A differential equation is solved if an equivalent equation is found involving only the independent and dependent variables.

This article has considered functions of a single independent variable only. Partial derivatives, multiple integrals, and partial differential equations are defined and studied in investigating functions of two or more independent variables.