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 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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): linear first-order differential equation