A systematic approach to obtain the analytical solution for linear second order ordinary differential equations: part I

The Leibniz integrating factor yields a reliable and direct way to solve linear first order ordinary differential equations. However, attempts to extend the technique to second or higher order differential equations resulted in integrating factors that are function of both the dependent and the independent variable and, thus, are the result of partial differential equations. Due to the importance of second and higher order ordinary differential equations in Applied Mathematics, Physics and Engineering, a simpler integrating factor, i.e., function of the independent variable only, gives important results to many problems that, today, rely on a multitude of different and more specific solution methods. Hence, this manuscript introduces the concept of generalized integrating factor for linear ordinary differential equations of order n. The procedure is used to address linear second order equations with varying and with constant coefficients. The solutions are analytically derived by means of nested convolutions and a close relation between linear and nonlinear differential equations is established. One interesting aspect of the proposed formulation is the fact that both the analytical homogeneous and the analytical particular solutions are obtained in separate. Analytical solutions for Bessel, Cauchy–Euler and the constant coefficients cases are provided and compared to examples in the literature and using numerical methods. Special attention is given to the constant coefficients case, since its application in mechanical and electrical engineering is widespread. Thus, a set of continuous excitation—periodic and polynomial—and discontinuous excitation functions—Dirac’s delta and Heaviside step function—are studied and their analytical solution are given. The analytical results were compared to each established method, like undetermined coefficients, variation of parameters and Laplace transform, showing the exactness and convenience of this generalization of the Leibniz integrating factor.