Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients

From ProofWiki
Jump to navigation Jump to search

Definition

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.


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 see

$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$
where $C$ is an arbitrary constant.


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


Sources