Construction of approximate solutions for a class of first-order nonlinear differential equations in the analyticity region

Nonlinear ordinary differential equations are mathematical models of various processes and phenomena of the real world; they belong to one of the most complicated categories of differential equations due to the presence of integrals of movable singular points. The study tested a class of first-order nonlinear ordinary differential equations with polynomial right part of not lower than the third degree. The solutions of these equations have movable singular points. The equations are not integratable in quadratures in a common case. For solving nonlinear differential equations with movable singular points of the algebraic type, we use the approximate method proposed by V.N. Orlov. We prove the existence and uniqueness of Cauchy problem solution for the class of differential equations in the analyticity region. In proving this theorem, we use the method of majorants for solving the examined nonlinear differential equations, rather than for solving the right part of differential equations, as it is done in classic literature. We offer the structure of approximate analytical solutions to the equations under study with the exact and perturbed values of the initial conditions, and we estimate the errors for these approximate solutions. The findings of the research are illustrated with the examples of calculations which are compared with similar results performed by other researchers.

References

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