Abstract
We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.
Highlights
The class of equations solved by elliptic functions, like (2.1) we first examine here (Section 2), triggered off the problem of the classification of the general nonlinear differential equation to special categories with respect to the character of the singular points of the solutions
We consider a method leading to a more general analytical solution for this specific equation, and in Section 4 we develop a general analytical technique applicable to another nonlinear ODE (equation (4.1)) not included in the above list of 50 equations. This technique is based on the introduction of a new function, called ᏼ, by which we obtain an exact closed form solution depending on the parameters of the equation and containing two arbitrary constants which yield a special solution in the case of an initial value problem
Evaluating hξ and hξξ by means of (3.1), introducing the results together with (3.1) into (2.5), and substituting (3.2) and (3.3), after some algebra, we obtain a polynomial of P in a more general form than that presented in (2.10), namely, 4(3 − n)ε23anPn+4 + (24 − 10n)ε22ε1anPn+3 + · · · = 0
Summary
The class of equations solved by elliptic functions, like (2.1) we first examine here (Section 2), triggered off the problem of the classification of the general nonlinear differential equation to special categories with respect to the character of the singular points of the solutions. This technique is based on the introduction of a new function, called ᏼ, by which we obtain an exact closed form solution depending on the parameters of the equation and containing two arbitrary constants which yield a special solution in the case of an initial value problem.
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