Variational calculus is used to determine the integrability of differential equations. A remarkable unified approach is presented by a single theorem employing variational calculus to determine the integrability of any ordinary differential equation whether linear or nonlinear. The theorem is also used to determine the integrating factor for a given equation if it is not directly integrable. Well established results for determining integrating factors obtained by various methods can be combined in a single equation of variational calculus. Many sample problems are extensively treated to show the power and applicability of the theorem. The method is applied to a variety of problems stemming from physical phenomena. Manuscript profile

A new numerical algorithm is proposed for solving ordinary differential equations. The algorithm is names as Residue Annihilation Method (RAM). The method does not require transfer of the equation into a first order system of equations. For a k’th order nonlinear ordinary differential equation, a parametric solution containing k+2 parameters is assumed as an initial step. By imposing the compatibility conditions together with the annihilation of the residue and its first derivative, a nonlinear system with k+2 equations is obtained. Solving the system yields a recursive relation for the parameters. The assumed parameter values therefore vary at each integration step. Evaluating the parametric solution at each integration step yields the discrete numerical solution. A continuous approximate solution valid throughout the whole domain can also be expressed in terms of the Gamma Interval Functions. Sample ordinary differential equations up to third order derivatives are treated with the new method. The method can be applied to initial value problems directly and to boundary value problems when combined with shooting techniques. Depending on the assumed parametric solution, better convergence can be achieved to the real solution. The convergence rate for the algorithm is O(h2), h being the step size. By including higher order derivatives of the residue, convergence rate can be increased. Manuscript profile