Editing Linear differential equation (section)

==Basic terminology==

The highest [[order of derivation]] that appears in a (linear) differential equation is the ''order'' of the equation. The term {{math|''b''(''x'')}}, which does not depend on the unknown function and its derivatives, is sometimes called the ''constant term'' of the equation (by analogy with [[algebraic equation]]s), even when this term is a non-constant function. If the constant term is the [[zero function]], then the differential equation is said to be ''[[Homogeneous differential equation|homogeneous]]'', as it is a [[homogeneous polynomial]] in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the ''{{visible anchor|associated homogeneous equation}}''. A differential equation has ''constant coefficients'' if only [[constant function]]s appear as coefficients in the associated homogeneous equation.

A ''{{visible anchor|solution|Solution of a differential equation}}'' of a differential equation is a function that satisfies the equation.
The solutions of a homogeneous linear differential equation form a [[vector space]]. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.