ABSTRACT
This chapter is devoted to the detailed study of second order differential equations with constant coefficients. The methodology based on the roots of the characteristic equation has been the primary focus for the determination of the solution. Unlike the conventional pedagogic formats where other approaches for obtaining the solutions are presented separately either in the Appendix or in other chapters, all the methods to obtain the solutions are presented together. These methods include breaking the second order differential equation into a pair of coupled first order differential equations, Laplace transforms, as well as numerical methods based on Runge-Kutta using Matlab. This allows the reader to compare the results from different approaches. The results also include detailed analysis of the phase portraits which help explain the stability of the systems modeled through these differential equations. While homogeneous differential equations are solved through the multiple ways described above, non-homogeneous differential equations are solved through the use of method of variation of parameters as well as the method of undetermined coefficients in order to obtain the particular solution. The results from these two approaches for obtaining the particular solutions are compared and explanations are provided appropriately in cases where the particular solutions appear different. Even in the case of the non-homogeneous differential equations, Laplace transforms as well as ODE based methods are employed as additional verification steps. The examples cover a substantial number of different cases and the solution in each case is prepared to be self-contained with appropriate theory.
