System of First Order Linear Homogeneous Differential Equations (FLHE) and Riccati Equation

Abstract

We present a simple method of solving a system of two first order linear homogeneous differential equations with variable co-efficients and demonstrate a correspondence of the system of equations with non-linear Riccati equation. By using appropriate substitution, this non-linear equation is converted into second order linear homogeneous differential equation, the solution of which is used to find the solution of the system of equations. It is seen that different sets of the variable co-efficients of the system of equations correspond to second order linear homogeneous differential equations of different nature. We have illustrated the method with one example. We have shown that the method can be applied to obtain the approximate analytical solutions of the spin dependent coupled Altarelli–Parisi evolution equation in Quantum Chromo Dynamics under plausible conditions.