Definition:Differential Equation/Linear

Definition

A linear differential equation is a differential equation where all dependent variables and their derivatives appear to the first power.

Neither are products of dependent variables allowed.

Hence a linear differential equation is of the form:

$\map {P_0} x y + \map {P_1} x \dfrac {\d y} {\d x} + \map {P_2} x \dfrac {\d^2 y} {\d x^2} + \cdots + \map {P_n} x \dfrac {\d^n y} {\d x^n} = \map Q x$

where $P_0, P_1, \ldots, P_n, Q$ are functions of $x$.

Examples

Arbitrary Example

The following is an example of a linear differential equation:

$x \dfrac {\d y} {\d x} + y = \sin x$

Also see

• Definition:Non-Linear Differential Equation

• Results about linear differential equations can be found here.

Sources

• 1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions
• 1960: D.R. Bland: Vibrating Strings ... (previous) ... (next): Chapter $1$: Strings of Finite Length: $1.1$ Introduction
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.10$: Linear Equations
• 1977: A.J.M. Spencer: Engineering Mathematics: Volume $\text { I }$ ... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction: Classification of Differential Equations
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): linear differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): linear differential equation