[Paper Review] On dual equation in theory of the second order ODE's

Valerii Dryuma|ArXiv.org|Jan 22, 2007

Differential Equations and Numerical Methods2 references3 citations

TL;DR

This paper establishes a duality between second-order nonlinear ODEs of the form $ y'' + a_1(x,y)y'^3 + 3a_2 y'^2 + 3a_3 y' + a_4 = 0 $ and dual equations $ b'' = g(a,b,b') $, where $ g $ satisfies a specific nonlinear fourth-order PDE. The key contribution is deriving and solving this PDE via parametric reduction and transformation methods, leading to explicit solutions involving the Lambert W function and Abel-type equations.

ABSTRACT

The relations between the second order ODE's cubical on the first derivative and their dual equations are discussed

Motivation & Objective

• To establish a geometric and algebraic duality between two classes of second-order nonlinear ODEs.
• To derive the nonlinear PDE satisfied by the dual function $ g(a,b,b') $, linking the original ODE to its dual.
• To develop a method for solving the PDE using parametric transformations and reductions.
• To construct explicit solutions for the dual equation through symmetry reductions and special function identities.
• To demonstrate the integrability of the system via the Legendre transformation and reduction to Abel-type equations.

Proposed method

• Derive the dual equation $ b'' = g(a,b,b') $ from the original ODE by analyzing curvature forms in the space of linear elements.
• Use a parametric representation $ f(x,y,z) \to u(x,t,z), y \to v(x,t,z) $ to reduce the high-order PDE (3) into a system involving $ u, v $ and their partial derivatives.
• Apply reductions of the form $ g = c^\alpha \omega[ac^{\alpha-1}] $, $ g = b^{1-2\alpha}\omega[cb^{\alpha-1}] $, etc., to simplify the PDE into solvable forms.
• Integrate the reduced PDEs using transformations such as the Legendre transformation and substitution $ \phi(x,y,z) = A(y/z) x^{-1} $.
• Solve the resulting ODEs for $ A(\eta) $ by reducing them to Abel-type equations and expressing solutions via the Lambert W function.
• Verify solutions by substitution into the original PDE and confirm consistency with the dual ODE structure.

Experimental results

Research questions

• RQ1 What PDE must the function $ g(a,b,b') $ satisfy for the dual equation $ b'' = g(a,b,b') $ to be equivalent to a second-order ODE of the form $ y'' + a_1 y'^3 + \cdots + a_4 = 0 $?
• RQ2 How can the nonlinear PDE (3) for $ g $ be systematically reduced and solved using parametric transformations and symmetry assumptions?
• RQ3 What is the role of the function $ h(a,b,c) $ in simplifying the PDE, and how does setting $ h=0 $ lead to integrable cases?
• RQ4 Can solutions of the dual equation be constructed explicitly using special functions like the Lambert W function?
• RQ5 What is the connection between the dual ODE and the Abel equation, and how does this enable explicit integration?

Key findings

• The dual function $ g(a,b,b') $ satisfies a specific fourth-order nonlinear PDE (3), which is derived from Cartan's curvature forms in the space of linear elements.
• The PDE (3) admits reductions such as $ g = c^\alpha \omega[ac^{\alpha-1}] $, allowing systematic construction of solutions.
• A solution to the reduced PDE is found in terms of the Lambert W function: $ A(\eta) = \text{LambertW}(\cdots) \cdot C_3 / (\text{LambertW}(\cdots) + 1) $, with $ \eta = y/z $.
• The dual ODE $ b'' = g(a,b,b') $ is shown to be cubic in $ b' $, confirming its nonlinear structure.
• The general integral of the system is expressed as $ y - (-a/(x-a))^{-4a} 16^x (-x/(x-a))^{4x} b (-2x+2a)^{-4a} = 0 $, providing a complete solution family.
• The solution process reduces to solving an Abel-type equation: $ db/da = -a(2a^2 + 7a + 6)b^3 + (-3a - 7)b^2 - 3b/a $, with elementary particular solutions.