Definition:Cauchy-Euler Equation

Definition

The linear second order ordinary differential equation:

$x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$

is the Cauchy-Euler equation.

General Form

Let $n \in \Z_{>0}$ be a strictly positive integer.

The linear ordinary differential equation:

$a_n x^n \, \map {y^{\paren n} } x + \dotsb + a_1 x \, \map {y'} x + a_0 \, \map y x = 0$

is the $n$th order Cauchy-Euler equation.

Also known as

The Cauchy-Euler equation is also known as:

• Euler's equation
• The Euler-Cauchy equation
• Euler's (or Cauchy's) equidimensional equation.

Also see

• Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE

Source of Name

This entry was named for Augustin Louis Cauchy and Leonhard Paul Euler.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 18$: Basic Differential Equations and Solutions: $18.9$: Euler or Cauchy Equation
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $4$