[Paper Review] Exact Solution of Abel Differential Equation with Arbitrary Nonlinear Coefficients

Ali Bakhshandeh Rostami|arXiv (Cornell University)|Mar 18, 2015

Numerical methods for differential equations4 citations

TL;DR

This paper presents an exact analytical solution for the Abel differential equation with arbitrary nonlinear coefficients, a class previously considered unsolvable. The method employs a novel transformation and integration technique applicable to any nonlinear form, verified via exact and numerical solutions against known cases, demonstrating high reliability and generality in solving this longstanding problem in nonlinear ODEs.

ABSTRACT

This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any arbitrary form of nonlinear coefficients of Abel equation. On the other hands, the presented solution in this paper is verified with exact solution for some restricted forms of Abel equation that has been reported by Polyanin and Zaytsev [5]. Also, numerical solution is utilized to verify this method. All verifications have approved that the present method is strongly reliable and exact to solve Abel equation, analytically.

Motivation & Objective

• To provide an exact analytical solution for the Abel differential equation, which has long been considered unsolvable in its general form.
• To develop a method applicable to any arbitrary nonlinear coefficient structure in the Abel equation.
• To verify the proposed solution against known exact solutions from Polyanin and Zaytsev for restricted forms of the equation.
• To demonstrate the method's reliability through numerical validation.
• To establish a general framework for solving nonlinear ODEs of the Abel type without restrictive assumptions.

Proposed method

• A novel transformation is introduced to reduce the general Abel equation to a solvable form using an integrating factor approach.
• The method relies on a specific substitution that linearizes the structure of the equation under certain conditions.
• The solution is derived through direct integration after transformation, yielding a closed-form expression.
• The approach is generalized to handle arbitrary nonlinear coefficients without requiring special functional forms.
• Numerical simulations are performed to validate the analytical results across various coefficient configurations.
• The method is tested against known exact solutions from the literature, ensuring consistency and correctness.

Experimental results

Research questions

• RQ1 Can an exact analytical solution be derived for the Abel differential equation with arbitrary nonlinear coefficients?
• RQ2 Does the proposed method maintain accuracy and consistency across different forms of nonlinear coefficients?
• RQ3 How does the proposed solution compare with known exact solutions for restricted cases of the Abel equation?
• RQ4 To what extent does the method remain reliable when validated numerically?
• RQ5 Can the solution be generalized to all forms of nonlinear coefficients without prior constraints?

Key findings

• The paper successfully derives an exact analytical solution for the general Abel differential equation with arbitrary nonlinear coefficients, resolving a long-standing problem.
• The solution is validated against known exact solutions from Polyanin and Zaytsev, showing perfect agreement for restricted cases.
• Numerical simulations confirm the method's reliability and accuracy across diverse coefficient configurations.
• The proposed method is applicable to any arbitrary form of nonlinear coefficients, demonstrating broad generality.
• The transformation and integration technique used in the solution process are robust and yield closed-form results without approximation.
• The results confirm that the Abel equation, previously considered unsolvable in full generality, admits an exact solution under the proposed framework.