# Definition:First Order Ordinary Differential Equation

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## 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:

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

## 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: $(1)$