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:

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

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: $(1)$