The paper presents a comparative analysis of the Differential Transform Method (DTM) with respect to classical numerical approaches, such as the Euler and Runge–Kutta methods, using the nonlinear Duffing equation as a representative example. This equation, being a prototypical model of dynamical systems exhibiting chaotic behavior, provides a demanding test environment for techniques used to approximate solutions of ordinary differential equations. The aim of the study is to assess the accuracy, stability, and computational efficiency of the considered methods as functions of the system parameters. The DTM approach, based on differential transform and a series representation of the solution, was compared with classical discretization schemes. In the DTM framework, the symmetry of the system is not imposed explicitly, but emerges from the initial conditions and the recursive structure used to determine the series coefficients. However, the preservation of this symmetry may be disrupted, leading to asymmetry due to truncation of the series expansion and the propagation of numerical errors. The obtained results indicate that DTM can serve as a competitive alternative to conventional methods, particularly in short-term simulations of nonlinear dynamical systems, offering high accuracy at a relatively low computational cost.

The first part of the paper presents a unique reduction of some general Riccati-type differential equation to the general Bernoulli-type equation. Furthermore four special theorems on reduction of some Riccati-type differential equations are considered in the second part of the paper. A number of examples are also presented to illustrate the considered issues.

Ramsey’s theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph $K_N$ contains either a red $K_n$ or blue $K_m$ as a subgraph. The smallest such N is called the Ramsey number, denoted as $R(n,m)$. In this paper, we reformulate this theorem and present a proof of Ramsey’s theorem that is novel as far as we are aware.