Solution of Differential Equations using Exponential of a Matrix

Abstract

Our main purpose in this project is to help reader find a clear and glaring relationship between linear algebra and differential equations, such that the applications of the former may solve the system of the latter using exponential of a matrix. Applications to linear differential equations on account of eigen values and eigenvectors, diagonalization of n-square matrix using computation of an exponential of a matrix using results and ideas from elementary studies form the core study of our project. Keywords : matrix,fundamental matrix, ordinary differential equations, systems of ordinary differential equations, eigenvalues and eigenvectors of a matrix, diagonalisation of a matrix, nilpotent matrix, exponential of a matrix I. Introduction The study of Ordinary Differential Equation plays an important role in our life. Some applications include study of growth of microorganisms, population, decay of radiation, etc. Ordinary Differential equations is also used in medicine. Solving a first order Ordinary Differential Equation of first degree could be elementary as we have many ways of doing so - the Ordinary Differential Equation could be linear, homogenous; or we could solve it finding suitable integrating factor to make it exact, etc. In solving a second order non-homogenous differential equation, we have many methods namely: method of undetermined coefficient also called method of judicial guessing, method of variation of parameters, Inverse D-opertor method, etc. The homogenous part can well easily be solved by finding the roots of the auxiliary equation. However, as the order of the Ordinary Differential Equation goes higher, it becomes more tedious to solve the homogenous/non-homogenous part. In such cases, we reduce the n th order Ordinary Differential Equation into a system of n first order Linear Differential Equation.