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:
 * $\dfrac {\mathrm d y} {\mathrm d x} + P \left({x}\right) y = Q \left({x}\right)$

It is:


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


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


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

Its general solution is:
 * $\displaystyle y = e^{-\int P \ \mathrm d x} \left({\int Q e^{\int P \ \mathrm d x}\ \mathrm d x + C}\right)$

where $C$ is an arbitrary constant.

Also see

 * Solution to Linear First Order Ordinary Differential Equation.