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On local symbolic approximation and resolution of ODEs using Implicit Function Theorem

Gianluca Argentini

TL;DR

The paper addresses local symbolic resolution of ODEs by leveraging the Implicit Function Theorem (and Dini's Theorem) to rewrite equations $F(x,y,y')=0$ in local normal forms $y'(x)=\phi(x,y)$ (or $y(x)=\psi(x,y'))$, enabling analytical local approximations. By expanding $\phi$ (and $\psi$) via Taylor series, it derives first- and higher-order local approximations that produce explicit, computable local solutions, with examples illustrating the method's accuracy and practicality even when global solutions are unavailable or non-elementary. A key insight is the observed connection between these implicit-function approximations and the standard series expansion of the solution, revealing that the implicit approach can recover known local behavior and offer tractable reduced equations for analysis. The techniques have practical value for obtaining analytical dependence on parameters and for informing symbolic forms of possible solutions in physical and engineering contexts, as demonstrated in cavitation and other nonlinear ODEs.

Abstract

In this work the implicit function theorem is used for searching local symbolic resolution of differential equations. General results of existence for first order equations are proven and some examples, one relative to cavitation in a fluid, are developed. These examples seem to show that local approximation of non linear differential equations can give useful informations about symbolic form of possible solutions, and in the case a global solution is known, locally the accuracy of approximation can be good.

On local symbolic approximation and resolution of ODEs using Implicit Function Theorem

TL;DR

The paper addresses local symbolic resolution of ODEs by leveraging the Implicit Function Theorem (and Dini's Theorem) to rewrite equations in local normal forms (or , enabling analytical local approximations. By expanding (and ) via Taylor series, it derives first- and higher-order local approximations that produce explicit, computable local solutions, with examples illustrating the method's accuracy and practicality even when global solutions are unavailable or non-elementary. A key insight is the observed connection between these implicit-function approximations and the standard series expansion of the solution, revealing that the implicit approach can recover known local behavior and offer tractable reduced equations for analysis. The techniques have practical value for obtaining analytical dependence on parameters and for informing symbolic forms of possible solutions in physical and engineering contexts, as demonstrated in cavitation and other nonlinear ODEs.

Abstract

In this work the implicit function theorem is used for searching local symbolic resolution of differential equations. General results of existence for first order equations are proven and some examples, one relative to cavitation in a fluid, are developed. These examples seem to show that local approximation of non linear differential equations can give useful informations about symbolic form of possible solutions, and in the case a global solution is known, locally the accuracy of approximation can be good.

Paper Structure

This paper contains 6 sections, 4 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ODEs and Dini's Theorem
  3. Local symbolic approximation for an ODE
  4. Relationship with series expansion of solution
  5. Finer examples
  6. Technical notes

Key Result

Theorem 1

Let $F(x,y(x),y'(x))=0$ an ODE, with $F \in C^1\left([a,b] \times \mathbb{R}^2\right)$ a real function $F(x,p,q)$, and let $(x_0,p_0,q_0)$ such that $x_0 \in (a,b)$, $F(x_0,p_0,q_0)=0$ and $\partial_q F(x_0,p_0,q_0) \neq 0$. Then exist $\delta > 0$, $\rho > 0$, a function $u \in C^1([x_0-\delta,x_0+

Figures (2)

  • Figure 1: Comparison between exact and approximated solution in the interval $\left[\frac{\pi}{3}-1,\frac{\pi}{3}+1\right]$; the exact solution is dashed.
  • Figure 2: Comparison between symbolic approximated solution of the reduce ODE (continuous line) and numerical solution of the original ODE (dashed line); case $p_{f}=1$, $\rho=1$, $R_0=0.1$; time is in horizontal axis, radius in vertical axis.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2