# Definition:Linear First Order Ordinary Differential Equation

## 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:

$\map Q x$ is a function of $x$
$a$ is a constant.

## 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 see

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

where $C$ is an arbitrary constant.

• Results about linear first order ODEs can be found here.