  We study modular differential equations for the basic weak Jacobi forms in
one abelian variable with applications to the elliptic genus of Calabi--Yau
varieties. We show that the elliptic genus of any $CY_3$ satisfies a
differential equation of degree one with respect to the heat operator. For a
$K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We
prove that for a general $CY_4$ its elliptic genus satisfies a modular
differential equation of degree $5$. We give examples of differential equations
of degree two with respect to the heat operator similar to the Kaneko--Zagier
equation for modular forms in one variable. We find modular differential
equations of Kaneko--Zagier type of degree $2$ or $3$ for the second, third and
fourth powers of the Jacobi theta-series.
