# Definition:Second Order Ordinary Differential Equation

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## Definition

A **second order ordinary differential equation** is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $2$.

The general **second order ODE** can be written as:

- $\displaystyle F \left({x, y, \frac {\mathrm d y} {\mathrm d x}, \frac {\mathrm d^2 y} {\mathrm d x^2}}\right)$

or, using prime notation:

- $F \left({x, y, y^{\prime}, y^{\prime \prime}}\right)$

## Historical Note

Much of the theory of Second Order ODEs was progressed by Leonhard Paul Euler.