**Keywords:**Darboux-Halphen-Ramanujan vector field | Hodge structure | Picard-Fuchs equation | Gauss-Manin connection

This research was intended as an attempt to obtain an ordinary

differential equation $\rm H$ from a linear differential equation

$L$. We work on a Calabi-Yau $n$-fold $W$ whose complex deformation is

one dimensional and middle complex de Rham cohomology

$H^n_{\rm dR}(W;\mathbb{C})$ is $(n+1)$-dimensional. Moreover, we suppose

that the Picard-Fuchs equation $L$ associated with the unique

nowhere vanishing holomorphic $n$-form $\omega$ on $W$ is of

order $n+1$. As a first result, we prove that $L$ is self-dual.

Next, we define ${\rm \textsf{T} }$ to be the moduli of $W$ together with a

basis of $H^n_{\rm dR}(W;\mathbb{C})$, where the basis is required to be

compatible with the Hodge filtration of $H^n_{\rm dR}(W;\mathbb{C})$;

furthermore, the intersection form matrix in this basis is

constant. Then in our second result, we verify the existence of

a unique vector field $\rm H$ on ${\rm \textsf{T} }$ that satisfies certain

properties. Indeed, the ordinary differential equation given by

$\rm H$ is a generalization of differential

equations introduced by Darboux, Halphen and Ramanujan.