Definition:First Order Ordinary Differential Equation

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A first order ordinary differential equation is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $1$.

The general first order ODE can be written as:

$\map F {x, y, \dfrac {\d y} {\d x} }$

or, using prime notation:

$\map F {x, y, y'}$

If it is possible to do so, then it is often convenient to present such an equation in the form:

$\dfrac {\d y} {\d x} = \map f {x, y}$

that is:

$y' = \map f {x, y}$

It can also be seen presented in the form:

$\map \phi {x, y, y'} = 0$

Also known as

A first order ordinary differential equation is often seen referred to just as a first order differential equation by sources which are not concerned about partial differential equations.

Some sources hyphenate: first-order differential equation.

The abbreviation ODE is frequently seen, hence first order ODE for first order ordinary differential equation.

Also see

  • Results about first order ODEs can be found here.