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