Linearization of second order ordinary differential equations via Cartan's equivalence method

Grissom, Charles B.; Thompson, Gerard; Wilkens, George R.

doi:10.1016/0022-0396(89)90154-x

Cited by 54 publications

(60 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We point out that the conditions (1.8) can also be deduced via the Cartan equivalence approach [7]. More recently in two independent papers [8,9], the authors present geometrical proofs for the determination of the invariant conditions (1.8).…”

Section: 2)mentioning

confidence: 90%

An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

Al-Dweik¹,

Mustafa²,

Mara'Beh³

et al. 2015

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between λ-symmetries and first integrals.This relation further leads to a new λ-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form y ′′ + F 1 (x, y)y ′ + F (x, y) = 0. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.

show abstract

Section: 2)mentioning

confidence: 90%

An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

Al-Dweik¹,

Mustafa²,

Mara'Beh³

et al. 2015

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…In the former reference the constraints for the coefficients of a linear ode that are necessary and sufficient for a maximal symmetry group of size n + 4 to exist are given explicitly up to order n = 9, see Eq. (3.19) of [3], page 90-91.…”

Section: Dye + Bdy -Ad X + (By -2ax)d + Gb~x-gc~y + Gc(b~-2cy) =0mentioning

confidence: 96%

“…For the general n th order linear equation Mahomed and Leach [3] and Krause and Michel [4] have solved the symmetry problem completely. In the former reference the constraints for the coefficients of a linear ode that are necessary and sufficient for a maximal symmetry group of size n + 4 to exist are given explicitly up to order n = 9, see Eq.…”

Section: Dye + Bdy -Ad X + (By -2ax)d + Gb~x-gc~y + Gc(b~-2cy) =0mentioning

confidence: 99%

On 3rd order ordinary differential equations with maximal symmetry group

Schwarz¹

1996

Computing

View full text Add to dashboard Cite

Abstract--ZusammenfassungOn 3 ra Order Ordinary Differential Equations with Maximal Symmetry Group. The following theorem is proved by investigating the Janet bases of determining systems: In order that a 3 rd order quasilinear ordinary differential equation has a seven-parameter symmetry group it must have a certain structure, and a set of necessary and sufficient conditions for its coefficients must be satisfied. This theorem generalizes similar results for linear equations and for quasilinear equations of 2 na order. It is shown how this theorem facilitates the computation of closed form solutions. AMS Subject Classifications: 34A05, 34A25, 68Q40.Key words: Differential equations, symmetries, equivalence problem.Uber gewiihnliche Differentialgleichungen 3. Ordnung mit maximaler Symmetrie~7"uppe. Es wird der folgende Satz bewiesen mit Hiffe der Eigenschaften yon Janet Basen: Damit eine quasilineare gewdhnliche Differentialgleichung 3 ter Ordnung eine sieben-parametrige Symmetriegmppe hat, muB sic eine bestimmte Skxuktur haben und eine Menge notwendiger und hinreichender Bedingungen fiir die Koeffizienten muB erffdlt sein. Dieser Satz verallgemeinert ~aliche Ergebnisse fiir lineare Gleichungen und ffir quasilineare Gleichungen 2 ter Ordnung. Es wird gezeigt, wie dieser Satz die Bestimmung geschlossener Ldsungen erleichtert.

show abstract

“…Also, we find the general solution of (17) in terms of local variables. Using the moving coframe method, we obtain the structure equations for the symmetry pseudo-group of equation (18) in the form…”

Section: Linearizability Of the Generalized Hunter-saxton Equationmentioning

confidence: 99%

Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches

Morozov¹

2005

SIGMA

View full text Add to dashboard Cite

show abstract

Linearization of second order ordinary differential equations via Cartan's equivalence method

Cited by 54 publications

References 5 publications

An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

An alternative proof of Lie’s linearization theorem using a new λ-symmetry criterion

On 3rd order ordinary differential equations with maximal symmetry group

Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches

Contact Info

Product

Resources

About