Definition:Cauchy-Euler Equation
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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
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 $4$