What is a differential equation?

A differential equation is an equation that contains a derivative of an unknown function, such as dy/dx = 2x or y″ + y = 0. Solving it means finding the function y, not a number.

They describe anything that changes: a population growing, a hot cup cooling, money earning interest. The rule for the change is known; the differential equation helps find the quantity itself.

Two labels describe each equation:

• Order: the highest derivative in it. y′ is first order, y″ second order, and so on.
• Degree: the power of that highest derivative, once the equation is written as a polynomial in the derivatives.

Examples:

In the second example, (dy/dx)² does not decide the degree: only the power of the highest derivative (here d²y/dx², power 1) counts.

Degree not defined: if a derivative sits inside sin, cos, log or an exponential, as in y′ + e^(y′) = 0 or y″ + sin(y′) = 0, the equation is not a polynomial in its derivatives, so the degree is not defined. The order still is.

But sin y or eˣ on their own are fine: they contain no derivative. The rule only looks at the derivatives.

Order and degree, when they exist, are always positive whole numbers.