  We investigate generating functions for the integrals over world-sheet tori
appearing in closed-string one-loop amplitudes of bosonic, heterotic and
type-II theories. These closed-string integrals are shown to obey homogeneous
and linear differential equations in the modular parameter of the torus. We
spell out the first-order Cauchy-Riemann and second-order Laplace equations for
the generating functions for any number of external states. The low-energy
expansion of such torus integrals introduces infinite families of
non-holomorphic modular forms known as modular graph forms. Our results
generate homogeneous first- and second-order differential equations for
arbitrary such modular graph forms and can be viewed as a step towards
all-order low-energy expansions of closed-string integrals.
