Reducing Reducible Linear Ordinary Differential Equations with Function Coefficients to Linear Ordinary Differential Equations with Constant Coefficients

In this article, I propose a generalized method for obtaining a substitution for reducing a reducible linear ordinary differential equation with function coefficients (RLDEF) to a linear ordinary differential equation with constant coefficient (LDE). This proposed method was also used to obtain the already known substitutions for the Euler’s and Legendre’s homogeneous second order linear differential equation. The derived method is able to reduce quite a large number of RLDEF to LDE including the Euler’s and Legendre’s homogeneous second order linear differential equation. However, these RLDEF (homogeneous and inhomogeneous) must satisfy the condition for reducibility, which is also proposed before the substitution is derived. the condition for reducibility is based on the order of the differential equation. In this article, the condition for reducibility is presented for a second and third order LDEF. Keywords: reducibility, generalized, differential equations.