In this article I want to summarize the connection between D-modules and differential equations.
Let \(P\in D_X\) be a differential operator, and let \(M = D/DP\). Then
\[\begin{align*} \mathrm{Hom}_{D_X}(M, \mathcal{O}_X) &= \mathrm{Hom}_{D_X}(D/DP, \mathcal{O}_X) \\ &= \{ \varphi\in \mathrm{Hom}_{D_X}(D, \mathcal{O}_X) : \varphi(P) = 0 \} \\ &= \{ f = \varphi(1) \in \mathcal{O}_X : \varphi(P) = P\varphi(1) = Pf = 0 \} \\ &= \ker(P : \mathcal{O}_X\to \mathcal{O}_X). \end{align*}\]
we see that the dual module \(\mathrm{Hom}_D(M,\mathcal{O})\) can be seen as the solution space of the differential equation \(Pf= 0\).