Application of Variational Calculus to Integrability of Diferential Equations with Physical Applications

Variational calculus is used to determine the integrability of dierential equations. a remarkable unied approach is presented by a single theorem employing variational calculus to determine theintegrability of any ordinary dierential equation whether linear or nonlinear. the theorem is alsoused to determine the integrating factor for a given equation if it is not directly integrable. wellestablished 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 showthe power and applicability of the theorem. the method is applied to a variety of problems stemming from physical phenomena.