Modular linear differential equations are differential equations invariant under modular transformations.
They play important roles in the study of 2D conformal field theory, vertex operator algebras and modular forms.
For example, characters of lisse (C_2-cofinite) vertex algebras and more generally, those of quasi-lisse vertex algebras,
satisfy modular linear differential equations. Moreover, they have also used in the attempt to classify lisse vertex algebras
from their characters. In this talk, we study a certain family of modular linear differential equations of fourth order
and discuss which vertex operator algebras satisfy
the differential equations.