  An element $a$ in a Banach algebra $\mathcal{A}$ has g-Drazin inverse if
there exists $b\in \mathcal{A}$ such that $ab=ba, b=bab$ and $a-a^2b
  \in \mathcal{A}^{qnil}$. In this paper we find new explicit representations
of the g-Drazin inverse of the block operator matrix $\left(
  \begin{array}{cc}
  E&I
  F&0
  \end{array} \right)$. We thereby solve a wider kind of singular differential
equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of
second order linear differential equations, Linear $\&$ Multilinear Algebra,
14(1983), 195--198].
