Bernoulli's differential equation

In this paper we propose a generalization of the famous Bernoulli differential equation by introducing a class of first order non-linear ordinary differential equations, which we call generalized Bernoulli differential equation. We also provide a family of solutions for it.

American Journal of Applied Sciences, 2020

The work shows the q-deformation of Bernoulli's equation, qderivative and q-calculus are used to form a q-analogous of Bernoulli's equation. We introduce the theorem of q-Bernoulli's equation.

HI9O

JOURNAL OF EDUCATION AND SCIENCE, 2008

Journal of Humanitarian and Applied Sciences, 2022

This research article discusses the Adomian decomposition method that has been applied to solving second-order the nonlinear (linear) fractional differential equation for the Bernoulli equation with initial conditions. Firstly, the Bernoulli equation with fractional derivatives is transferred to a nonlinear (linear) fractional differential equation subject to initial conditions. Then it investigated the existence of approximate solutions to this type of initial value problem by applying Adomian decomposition technique. In view of the convergence of this method, some illustrative examples are included to demonstrate the proposed technique and show the efficiency of the presented method.

Thermal Science, 2019

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

Journal of Humanitarian and Applied Sciences, 2020

In this study, we present a second-order nonlinear equation with the nonlinearity of the Bernoulli type, which includes fractional order derivatives. We consider the numerical solution of the nonlinear equation using the Picard iteration method, the method seeks to examine the convergence of solutions of this type of equation. The resulting solution showed that the convergence could be increased at each iterate level. However, as the number of iterations increases, there is a rapid rate of convergence of the approximate solution to the analytic solution. All Results obtained with the classical Picard method on the equation were compared with the exact solution.

SN Applied Sciences

In this research paper, the authors derived the non-linear differential equations for certain hybrid special polynomials related to the Bernoulli polynomials. The families of non-linear differential equations arising from the generating functions of the Bernoulli-Euler and Bernoulli-Genocchi polynomials are derived. Further, these non-linear differential equations are used to derive certain identities and formulas for the Bernoulli-Euler and Bernoulli-Genocchi numbers. However, to provided an exception, a linear differential equation is derived from the generating function of the Genocchi-Euler polynomials.

International Journal of Applied Mathematical Research

Fractional differential equations are often seeming perplexing to solve. Therefore, finding comprehensive methods for solving them sounds of high importance. In this paper, a general method for solving second order fractional differential equations has been presented based on conformable fractional derivative. This method realizes on determining a general solution of homogeneous and a particular solution of a second order linear fractional differential equations. Furthermore, a general solution has been developed for fractional Euler’s equation. For more explanation of each part, some examples have been solved.