Definition:Differential Equation/Ordinary

Definition

An ordinary differential equation is a differential equation which has exactly one independent variable.

All the derivatives occurring in it are therefore ordinary.

The general ordinary differential equation of order $n$ is:

$\map f {x, y, \dfrac {\d x} {\d y}, \dfrac {\d^2 x} {\d y^2}, \ldots, \dfrac {\d^n x} {\d y^n} } = 0$

or, using the prime notation:

$\map f {x, y, y', y, \ldots, y^{\paren n} } = 0$

Also known as

The term ordinary differential equation is often presented in its abbreviated form ODE or O.D.E.

Also see

• Results about ordinary differential equations can be found here.

Sources

• 1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions
• 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $3$. Definitions
• 1956: E.L. Ince: Integration of Ordinary Differential Equations (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions
• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Definition $3.1$
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions
• 1977: A.J.M. Spencer: Engineering Mathematics: Volume $\text { I }$ ... (previous) ... (next): Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction: Classification of Differential Equations
• 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ordinary differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordinary differential equation