Some Properties of Approximate Solutions of Linear Differential Equations

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ''(x)+alpha u '(x)+beta u(x)=r(x), with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ''(x)+alpha u '(x)+beta u(x)=0. Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c is an element of R we can have a good approximate solution near c with very small error estimation.