Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following form P(x). ln x and the Characteristic Equation Has Distinct Real Roots

For n-th order linear differential equations with constant coefficients the problem of determining the general solution involves determining the general solution of the attached homogeneous equation and, on the other hand, determining a particular solution. This paper aims to analyze the conditions under which the general solution can be obtained for a n-th order linear differential equation with constant coefficients which have the free term in the following form P(x)^. ln x under the condition that the attached characteristic equation has real and distinct roots, meaning the order of multiplicity for each root is 1. For this we will use the method of variation of constants which leads us to a system whose solution we will gain by using the Upper incomplete Gamma function.