Cauchy–Euler equation

E239291

equidimensional equation linear differential equation ordinary differential equation second-order differential equation variable-coefficient differential equation

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.

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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