# Definition:First Order Ordinary Differential Equation

## Definition

A **first order ordinary differential equation** is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $1$.

The general **first order ODE** can be written as:

- $\map F {x, y, \dfrac {\d y} {\d x} }$

or, using prime notation:

- $\map F {x, y, y'}$

If it is possible to do so, then it is often convenient to present such an equation in the form:

- $\dfrac {\d y} {\d x} = \map f {x, y}$

that is:

- $y' = \map f {x, y}$

It can also be seen presented in the form:

- $\map \phi {x, y, y'} = 0$

## Also known as

A **first order ordinary differential equation** is often seen referred to just as a **first order differential equation** by sources which are not concerned about partial differential equations**.**

Some sources hyphenate: **first-order differential equation**.

The abbreviation **ODE** is frequently seen, hence **first order ODE** for **first order ordinary differential equation**.

## Also see

- Results about
**first order ODEs**can be found**here**.

## Sources

- 1962: J.C. Burkill:
*The Theory of Ordinary Differential Equations*(2nd ed.) ... (previous) ... (next): Chapter $\text I$: Existence of Solutions: $2$. Simple ideas about solutions - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions - 1978: Garrett Birkhoff and Gian-Carlo Rota:
*Ordinary Differential Equations*(3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction: $(1)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**first-order differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**first-order differential equation**