Nonlinear differential equations of the second, third and fourth order with exact solutions (Q450269)

This paper presents the exact solution of a class of nonlinear ordinary differential equations of second, third and fourth order in polynomial form. The class of equations is written as NEWLINE\[NEWLINEy_{n,z} = F(y, y_z, y_{n-1, z}),\eqno{(*)}NEWLINE\]NEWLINE where \(y_{n,z}\) is the \(n\)th derivative NEWLINE\[NEWLINEy_{n,z} = {d^n y \over d z^n},NEWLINE\]NEWLINE and \(F\) is a polynomial in terms of the function \(y\) and its derivatives \(y_z, y_{zz}\) and so on. The goal is to determine \(F\) such that \((*)\) has exact solutions in the form NEWLINE\[NEWLINEy = \sum_{n=0}^N A_n Q^n,NEWLINE\]NEWLINE where \(A_n\) are coefficients, \(N\) is an integer and \(Q(z)\) takes the form NEWLINE\[NEWLINEQ(z)={1 \over 1+\exp(z)}.NEWLINE\]NEWLINE The algorithm consists of four steps which are (1) construction of the general form of the nonlinear ordinary equation with the general solution having integer poles; (2) substitution of the derivatives for the function \(y(z)\) with respect to \(z\) and the expression for \(y(z)\) into the original equation (\(*\)); (3) finding of algebraic equations for the coefficients \(A_n\) and parameters of the equation (\(*\)); (4) solving the system of algebraic equations to find the coefficients \(A_n\).