General exact solutions of certain second-order homogeneous algebraic differential equations

The general exact solutions of the second-order homogeneous algebraic differential equation [GRAPHICS] is studied, where [GRAPHICS] and [GRAPHICS] are three known bounded linear operators on a Banach space [GRAPHICS] . If [GRAPHICS] such that [GRAPHICS] is invertible, an anti-triangular operator matrix [GRAPHICS] is defined. When [GRAPHICS] is Drazin invertible, a necessary and sufficient condition is obtained in terms of [GRAPHICS] and its Drazin inverse [GRAPHICS] under which [GRAPHICS] becomes a solution of (0.1). Specifically, a formula for [GRAPHICS] is derived under the assumptions that [GRAPHICS] is a Hilbert space, [GRAPHICS] and [GRAPHICS] are positive semi-definite operators on [GRAPHICS] with the summation [GRAPHICS] being positive definite.