ALGORITHM FOR CONSTRUCTING THE GENERAL SOLUTION OF THE BERNOULLI DIFFERENTIAL EQUATION

Abstract

The Bernoulli differential equation occupies an important position in the theory of ordinary differential equations due to its nonlinear structure and its reducibility to a linear differential equation through an appropriate transformation. The present study develops a systematic and rigorous algorithm for obtaining the general solution of the Bernoulli differential equation and examines its theoretical foundation, transformation properties, and structural characteristics. The work provides a detailed mathematical framework describing the reduction of nonlinear equations of Bernoulli type to linear equations by means of power transformations and investigates the conditions under which such transformations are valid. Special attention is given to the existence and uniqueness of solutions, exceptional cases, and structural properties of the resulting linear equations. The proposed algorithm is analyzed from both analytical and pedagogical perspectives, emphasizing its applicability in solving nonlinear problems encountered in applied mathematics, mathematical physics, engineering sciences, and mathematical modeling. The study demonstrates that the Bernoulli equation serves as an intermediate class between linear and nonlinear differential equations, illustrating fundamental methods of nonlinear transformation and linearization. The results establish a coherent mathematical procedure for constructing general solutions and clarify the theoretical significance of Bernoulli equations in the qualitative theory of differential equations.