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

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.