# Definition:Differential Equation/Degree

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## Definition

Let $f$ be a **differential equation** which can be expressed as a polynomial in all the derivatives involved.

The **degree** of $f$ is defined as being the power to which the derivative of the highest order is raised.

By default, if not specifically mentioned, the **degree** of a **differential equation** is assumed to be $1$.

### First Degree (Quasilinear)

A **quasilinear differential equation** is a differential equation of the **first degree**.

## Examples

### First Degree Second Order

- $\dfrac {\d^2 y} {\d x^2} = k x$

is an example of an **order $2$** ordinary differential equation of **degree $1$**.

### Second Degree Second Order

- $\paren {\dfrac {\d^2 y} {\d x^2} }^2 = k x$

is an example of an **order $2$** ordinary differential equation of **degree $2$**.

## Also see

- Results about
**the degree of a differential equation**can be found**here**.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions - 1952: H.T.H. Piaggio:
*An Elementary Treatise on Differential Equations and their Applications*(revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $3$. Definitions - 1956: E.L. Ince:
*Integration of Ordinary Differential Equations*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**degree**:**3.**(of a differential equation) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**degree**:**3.**(of a differential equation) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**differential equation**