Cauchy–Euler equation

E239291

The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (2)

Label Occurrences
Cauchy–Euler equation canonical 2
Euler–Cauchy equation 1

Statements (48)

Predicate Object
instanceOf equidimensional equation
linear differential equation
ordinary differential equation
second-order differential equation
variable-coefficient differential equation
appearsIn classical ODE textbooks
hasAlternativeName Cauchy–Euler equation
surface form: Euler–Cauchy equation

equidimensional equation
hasChangeOfVariable t = ln x
x = e^t
hasCharacteristicEquation a m (m - 1) + b m + c = 0
hasCoefficient a x^2
b x
c
hasDependentVariable y(x)
hasDomainRestriction x < 0
x > 0
hasExtension nonhomogeneous Cauchy–Euler equation
hasGeneralForm a x^2 y'' + b x y' + c y = 0
hasGeneralNonhomogeneousForm a x^2 y'' + b x y' + c y = f(x)
hasIndependentVariable x
hasKeyConcept equidimensionality in x
reduction to constant coefficients via logarithmic substitution
hasOrder second order
hasProperty homogeneous
linear in y, y', y''
power-law type
scale invariant
singular at x = 0
hasSolutionForm y = C1 x^{m1} + C2 x^{m2} (distinct real roots)
y = C1 x^{m} + C2 x^{m} ln x (repeated root)
y = x^{ ho} [C1 \, cos(eta \, ln x) + C2 \, sin(eta \, ln x)] (complex roots)
hasStandardSolutionMethod power-law ansatz y = x^m
reduction to constant-coefficient ODE via logarithmic change of variables
hasType homogeneous linear ODE
isSpecialCaseOf linear ordinary differential equation with variable coefficients
isTaughtIn applied mathematics courses
undergraduate differential equations courses
isUsedIn elasticity theory
engineering differential equations
fluid mechanics
heat conduction in cylindrical coordinates
heat conduction in spherical coordinates
mathematical physics
problems with power-law behavior
problems with scale-invariant behavior
namedAfter Augustin-Louis Cauchy
Leonhard Euler

How these facts were elicited

The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.

Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10.

# Requirements
- If you don't know the subject at all, return an empty list.
- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
- Do not get too wordy.
- Separate several objects into multiple triples with one object.
Input
Subject: Cauchy–Euler equation
Description of subject: The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy knownFor Cauchy–Euler equation
Augustin-Louis notableFor Cauchy–Euler equation
subject surface form: Augustin-Louis Cauchy
Cauchy–Euler equation hasAlternativeName Cauchy–Euler equation
this entity surface form: Euler–Cauchy equation