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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy–Euler equation canonical | 2 |
| Euler–Cauchy equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171652 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
Target entity: Cauchy–Euler equation Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy–Euler equation]
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A.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
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B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
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C.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
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D.
ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
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E.
d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Cauchy–Euler equation Target entity description: 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.
-
A.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
-
B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
C.
Euler–Lagrange equation
The Euler–Lagrange equation is a fundamental differential equation in the calculus of variations that provides the condition for a function to make a functional stationary, forming the basis of Lagrangian mechanics and many physical theories.
-
D.
ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
-
E.
d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
- F. None of above. chosen
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.
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.
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.