# Definition:Linear First Order Ordinary Differential Equation

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## Definition

A **linear first order ordinary differential equation** is a differential equation which is in (or can be manipulated into) the form:

- $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

where $\map P x$ and $\map Q x$ are functions of $x$.

It is:

**Linear**because both $\dfrac {\d y} {\d x}$ and $y$ appear to the first power, and do not occur multiplied together

**First order**because the highest derivative is $\dfrac {\d y} {\d x}$

**Ordinary**because there are no partial derivatives occurring in it.

### Constant Coefficients

A **linear first order ordinary differential equation with constant coefficients** is a linear first order ordinary differential equation which is in (or can be manipulated into) the form:

- $\dfrac {\d y} {\d x} + a y = \map Q x$

where:

## Also presented as

A **linear first order ordinary differential equation** can also be presented as:

- $\dfrac {\d y} {\d x} = \map P x y + \map Q x$

or:

- $\dfrac {\d y} {\d x} + \map P x y + \map Q x = 0$

## Also known as

Some sources hyphenate: **linear first-order (ordinary) differential equation**.

## Also see

- Solution to Linear First Order Ordinary Differential Equation: Its general solution is:

- $\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$

where $C$ is an arbitrary constant.

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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 18$: Basic Differential Equations and Solutions: $18.2$: Linear first order equation - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.10$: Linear Equations - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**differential equation**: differential equations of the first order and first degree: $(5)$*Linear equations* - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**differential equation**: differential equations of the first order and first degree: $(5)$*Linear equations* - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**linear first-order differential equation**