# Euler equation

A linear ordinary differential equation of order $n$ of the form

$$\tag{1 } \sum _ { i= } 0 ^ { n } a _ {i} x ^ {i} \frac{d ^ {i} y }{d x ^ {i} } = f ( x ) ,$$

where $a _ {i}$, $i = 0 \dots n$, are constants and $a _ {n} \neq 0$. This equation was studied in detail by L. Euler, starting from 1740.

The change of the independent variable $x = e ^ {t}$ transforms (1) for $x > 0$ to the linear equation of order $n$ with constant coefficients

$$\sum _ { i= } 0 ^ { n } a _ {i} D ( D - 1 ) \dots ( D- i+ 1) y = f ( e ^ {t} ) ,\ \ D = \frac{d}{dt} .$$

The characteristic equation of the latter is called the indicial equation of the Euler equation (1). The point $x = 0$ is a regular singular point of the homogeneous Euler equation. A fundamental system of (real) solutions of the real homogeneous equation (1) on the semi-axis $x > 0$ consists of functions of the form

$$\tag{2 } x ^ \alpha \cos ( \beta \mathop{\rm ln} x ) \mathop{\rm ln} ^ {m} x ,\ \ x ^ \alpha \sin ( \beta \mathop{\rm ln} x ) \mathop{\rm ln} ^ {m} x .$$

If $x < 0$, then (1) requires the substitution $x = - e ^ {t}$, and in (2) $x$ is replaced by $| x |$.

A more general equation than (1) is the Lagrange equation

$$\sum _ { j= } 0 ^ { n } a _ {j} ( \alpha x + \beta ) ^ {j} y ^ {(} j) = f ( x) ,$$

where $\alpha$, $\beta$ and $a _ {j}$ are constants and $\alpha \neq 0$, $a _ {n} \neq 0$, which can also be reduced to a linear equation with constant coefficients by means of the substitution

$$\alpha x + \beta = e ^ {t} \ \ \textrm{ or } \ \alpha x + \beta = - e ^ {t} .$$

## Contents

How to Cite This Entry:
Euler equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_equation&oldid=46858
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article