Ordinary differential equations (ODEs) are the most powerful mathematical tools found in the physical sciences. Many topics in mathematics, physics, and engineering are related to linear and nonlinear ODEs or systems of ODEs. The series of problems in ODEs with initial and boundary conditions arises in experimental physics, numerical simulation approaches, and many other scientific fields, including engineering, signals processing, economics, and acoustics. Many researchers have used various techniques to find approximate solutions to systems of ODEs. For illustration, Mesady et al. [
1] presented a Jafari transformation for a system of linear ODEs with medical applications. By using the Jafari transform, ODEs can be transformed into a series of algebraic equations. Higazy and Aggarwal [
2] presented the Sawi transformation for systems of ODEs. Sawi transformations have been used to obtain the concentration of chemical reactions for different examples from physical chemistry. In [
3], Ricky et al. proposed the neural ordinary differential equations. This technique is based on parameterizing the derivative of the hidden state rather than specifying a deep series of concealed layers. In [
4], Brown and Biggs provided methods for solving the ODEs for unconstrained optimization, while Zadunaisky [
5] proposed a numerical technique to find the errors in the numerical solutions for the system of ODEs. In order to solve stiff and non-stiff systems of ODEs, in [
6] Linda presented automatic selection of methods, which provides a way of determining whether or not a problem can be addressed by a set of strategies suitable for stiff or non-stiff problems. In recent years, many methods based on numerical investigation of nonlinear moving boundary problems, temperature-dependent numerical studies, the solutions of the nonlinear regularized long wave equation, and the nonlinear sinh-Gordon model have been proposed to find numerical solutions of physical models [
7,
8,
9,
10,
11]. In [
12], Neuberger presented the steepest descent in Hilbert spaces for a general system of linear differential equations. To forecast traffic on a limited time scale based on evolutionary algorithms, Chen et al. [
13] explained time series forecasting using a system of ODEs. Farshid [
14] proposed a differential transform method for systems of ODEs. The main aim of this effort was to catch the exact solution when the solution is known in terms of series expansion. Biazer et al. [
15] presented the solution of a system of ODEs using the Adomian decomposition method by converting the given system into a system of first-order ODEs. Kurnaz and Galip [
16] presented a comprehensive discussion of solving a system of ODEs by adjusting the step size. Their method allows for control of the truncation error used in numerical methods. In view of the above literature, a very small number of attempts have been made to find the numerical solution of a system of ODEs, and there are no works as yet addressing ways to find the numerical solution of a fourth-order ODE on any finite interval
. In this paper, a new method for finding a numerical solution to a system of fourth-order ODEs on any given interval is developed. It is demonstrated that the suggested method works well and is suitable for solving a linear system of fourth-order ODEs. Additionally, while the Hyers–Ulam stability of ODEs has recently been studied [
17,
18,
19,
20], no stability analysis of a system of ODEs for any given arbitrary interval has yet been carried out. In this paper, a new method is presented for solving systems of linear and nonlinear fourth-order ODEs using operational matrices of generalised symmetric Bernstein polynomials. Furthermore, Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided for a system of fourth-order ODEs on any finite interval.