  For every positive integer $r$, we solve the modular Schwarzian differential
equation $\{h,\tau\}=2\pi^2r^2E_4$, where $E_4$ is the weight 4 Eisenstein
series, by means of equivariant functions on the upper half-plane. This paper
supplements previous works \cite{forum, ramanujan}, where the same equation has
been solved for infinite families of rational values of $r$. This also leads to
the solutions to the modular differential equation $y''+r^2\pi^2E_4\,y=0$ for
every positive integer $r$. These solutions are quasi-modular forms for
$\mbox{SL}_2(\mathbb Z)$ if $r$ is even or for the subgroup of index 2,
$\mbox{SL}_2(\mathbb Z)^2$, if $r$ is odd.
