Eulerian Method for Third Order Linear Systems

Abstract

In this thesis, we have studied the solutions of the various types of third order linear systems of ordinary differential equations. In chapter I, we have considered the third order linear homogeneous system with constant coefficients and developed Eulerian methods for all possible cases of the characteristic roots of the variational matrix. In chapter 2, we have considered the third order linear nonhomogeneous system with constant coefficients. The method covers all the cases when the roots of the characteristic equation of the corresponding homogeneous linear system are real and distinct, real and equal and complex. In finding particular solutions for the nonhomogeneous system of equations we have used the method of variation of parameters. Finally, we have obtained solutions of this system with the help of Crammer's rule. In chapter 3, we have considered the generalized form of third order linear nonhomogeneous system with constant coefficients. We have extended the Eulerian method and developed new techniques for obtaining solutions of this system. In chapter 4, we have discussed the third order linear nonhomogeneous system with variable coefficients. This problem is very difficult to solve, so we have examined a special case of this problem. By using a suitabletransformation, we have reduced it to a third order linear nonhomogeneous system with constant coefficients and have found solutions by the method of Chapter 2. We have illustrated all the methods by several suitable examples.