An analytical approach to solve boundary value problems for autonomous second order nonlinear differential equations

This paper focuses on two-point boundary value problems for autonomous second order nonlinear differential equations of the form y'' = f(y,y') which can represent many problems in physics and engineering. Analytical integration of such equations leads to an indefinite integral, which in most cases cannot be expressed by elementary functions. In addition, obtaining an analytical expression for the unknown constants of integration C₁ and C₂ depending on the boundary conditions is also difficult or impossible, especially when an indefinite non-elementary integral is involved. Due to these limitations the implementation of such an analytical solution is very limited. The present paper proposes an alternative analytical approach aiming at extending the usage of analytical solutions to such problems. The developed approach is based on the linearization of the right-hand side f(y) of the original nonlinear equation in the neighborhood of some value y* of the unknown solution y(x) where y* serves as a parameter in the solution y_L (x, y*) of the linearized equation. A special (non-differential) equation is obtained that allows to calculate y* for each x. The proposed approach allows to obtain a solution at any selected point x, without calculating the solution at other points (in contrast to explicit numerical methods), and without solving a system of high-order nonlinear equations (in contrast to implicit numerical methods). Examples of different physical problems are presented to validate the proposed approach and demonstrate its high precision. The excellent agreement with known analytical and numerical solutions, is quantitatively estimated by the absolute and relative errors calculated using the norms L¹ and L².