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Notes on Ordinary Conformal Differential Equations of order $α$ ($0<α\leq 1$)

Carlos E. Cadenas R

TL;DR

The paper addresses solving conformable ODEs of order $\alpha$ with $0<\alpha\leq 1$ by adapting classical techniques. It introduces and develops conformable analogs of separable, homogeneous, linear, Bernoulli, and exact equations, supported by integrating factors and variable changes, and illustrates these methods with representative examples. The main contributions are systematic definitions (e.g., $(\alpha)$-separable, $(n,\alpha)$-homogeneous, $(\alpha)$-linear, $(\alpha)$-Bernoulli, $(\alpha)$-exact) and explicit solution strategies that reproduce familiar ODE techniques in the conformable setting. This framework provides a practical, educational pathway for applying conformable calculus in engineering and applied sciences, while clarifying the computational steps through concrete examples and formulas.

Abstract

These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous equations, linear, Bernoulli and exact. Representative examples are presented in all cases. Emphasis is placed on the new definitions and notations.

Notes on Ordinary Conformal Differential Equations of order $α$ ($0<α\leq 1$)

TL;DR

The paper addresses solving conformable ODEs of order with by adapting classical techniques. It introduces and develops conformable analogs of separable, homogeneous, linear, Bernoulli, and exact equations, supported by integrating factors and variable changes, and illustrates these methods with representative examples. The main contributions are systematic definitions (e.g., -separable, -homogeneous, -linear, -Bernoulli, -exact) and explicit solution strategies that reproduce familiar ODE techniques in the conformable setting. This framework provides a practical, educational pathway for applying conformable calculus in engineering and applied sciences, while clarifying the computational steps through concrete examples and formulas.

Abstract

These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous equations, linear, Bernoulli and exact. Representative examples are presented in all cases. Emphasis is placed on the new definitions and notations.

Paper Structure

This paper contains 12 sections, 4 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. Conformable derivative and $\alpha-$integral
  3. Preliminaries
  4. $(\alpha)$-separable differential equations
  5. A particular case
  6. Differential equations $(n,\alpha)$-homogeneous
  7. A particular case
  8. Convertible to $(1,\alpha)$-homogeneous
  9. ($\alpha$)-linear differential equations
  10. ($\alpha$)-Bernoulli differential equations
  11. ($\alpha$)-exact differential equations
  12. Final Comments

Key Result

Theorem 2.2

Khalil2014 Let $\alpha \in(0,1]$ and $f,g$ be $\alpha$-differentiable at a point $x>0$. Then

Theorems & Definitions (19)

  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Theorem 3.2
  • Example 4.1
  • Example 4.2
  • ...and 9 more