Application of the Drazin Inverse on the Solution of the first-order linear singular differential equations

The Drazin Inverse is a generalized form of inverse that shares similar properties with a square matrix; hence, the Drazin inverse is only defined for a square matrix. The Drazin inverse has numerous applications in solving singular differential equations, Markov chains, and iterative methods in numerical analysis. In the case where the system ???????? = ???? involves a matrix ???? that is invertible, if matrix ???? is singular or non-invertible "Certainly! The corrected text is: "inverse, denote"e, the Drazin inverse is utilized to solve the system above. The research aim is to utilize the Drazin inverse in solving first-order singular linear differential equations. Let ???? and ???? be ???? × ???? matrices, ???? a vector-valued function. ???? and ???? may both be singular. The differential equation ????????’ + ???????? = ???? is examined using the theory of the Drazin inverse. ???? closed form expression for all solutions of the differential equation is provided when the equation has unique solutions for consistent initial conditions. This is a review article and the results show that to solve single linear differential equations, the inverse of Drazin is the best possible way to solve this type of differential equations.