Definition:First Order Ordinary Differential Equation
Definition
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.
Sources
- 1962: J.C. Burkill: The Theory of Ordinary Differential Equations (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Existence of Solutions: $2$. Simple ideas about solutions
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions
- 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction: $(1)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): first-order differential equation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): first-order differential equation