  A characterization of the minimal $\mathcal{W}$-algebras associated with the
Deligne exceptional series at level $-h^\vee/6$ is obtained by using
one-parameter family of modular linear differential equations of order $4$. In
particular, the characters of the Ramond-twisted modules of minimal
$\mathcal{W}$-algebras related to the Deligne exceptional series satisfy one of
these differential equations. In order to obtain the characterization, the
differential equations in the one parameter family which have solutions of "CFT
type" are classified, whose solutions are explicitly described.
