Study and Analysis of the Second Order Constant Coefficients and Cauchy-Euler Equations via Modified Conformable Operator

In this paper, we are concerned with a new modified conformable operator. Such an operator makes the study very easy in fractional calculus because it satisfies the most properties as the usual derivative and gives exact solutions. Furthermore, we will analyze and study the second-order fractional linear homogeneous differential equation with constant coefficients, which has two reasons for the importance of these types of differential equations. First of all, they often arise in applications. Second, it is relatively easy to find fundamental sets of solutions to these equations. In addition, we will also analyze the related fractional Cauchy–Euler type equation, which is used in various fields, physics, engineering, etc. Finally, as an application, we will illustrate the method on some numerical examples of the mentioned type of fractional differential equations.