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.

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

where $C$ is an arbitrary constant.