Definition:Differential Equation/Linear
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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
- 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