On the Group Analysis of Differential Equations on the Group SL(2, R)

Abstract

We get a method for constructing explicitly the exact solution of the initial value problem defined by the linear differential equation \(\dot{g}(t) = A(t)g(t), \; g(0)=E\), where \(g(t)\in SL(2,R)\) and A(t) is a polynomial matrix belonging to the Lie algebra sl(2, R) obeying some constraints. The analysis is carried out by using the nilpotent elements of sl(2, R) and Lie algebraic techniques. The solution is presented in the form of finite product of exponentials.