Solution of linear systems of differential equations with singular constant coefficients by the Drazin inverse of matrices

ABSTRACT

Let 𝐴 ,𝐵 be 𝑛×𝑛 matrices of complex numbers. Let 𝐺 a vector-valued function of the real variable 𝑡. 𝐴 and 𝐵 may both be singular, rank(𝐴) = 1, and the trac of 𝐴 is not equal zero. The linear system of differential equations 𝐴𝑥′(𝑡)+𝐵𝑥(𝑡)=𝐺(𝑡) is studied using the Drazin inverse 𝐴𝐷 of 𝐴, and a new matrix 𝐾∈ℂ𝒏×𝒏. In this paper, we obtain a new closed form for the general solution of the differential system when the system is tractable.