Given a connection $\nabla_{\partial_x}: \mathbb{C}[x]\rightarrow \mathbb{C}[x]$, we can view it as a $\mathbb{C}[x,\partial_{x}]$ module where $\partial_{x}$ acts on $\mathbb{C}[x]$ by $\partial_x f = \nabla_{\partial_x}f$. In fact, giving $\mathbb{C}[t]$ a $\mathbb{C}[x,\partial_{x}]$ module structure is the same as defining a connection in the same way.
Now, given a $P\in C[x,\partial_x]$, are there any relations between the two $\mathbb{C}[x,\partial_x]$ modules:
1.$\frac{\mathbb{C}[x,\partial_x]}{\mathbb{C}[x,\partial_x]\cdot P}$
2.The $\mathbb{C}[x,\partial_x]$ module correspond to the connection $\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ where $\nabla_{\partial_x} f =P(f)$.
It seemes that they are not the same things. For example take $P=\partial_x$. The first is isomorphic to $\mathbb{C}[x]$ and $\partial_x$ acts by zero. The second is also $\mathbb{C}[x]$ where $\partial_x$ acts by $P=\partial_x$.
But they do share some relations. Applying the solution functor to the first give us $\{f\in \mathbb{C}[x] |Pf=0\}$ and the global horizontal sections of the second connection is also $\{f\in \mathbb{C}[x] |Pf=0\}$.
I find where the confusion comes: as a $\mathbb{C}[x,\partial_x]$ module the connection $\nabla = d+A$ should be $\mathbb{C}[x,\partial]/(\partial_x-A)$, rather than$\mathbb{C}[x,\partial]/(\partial_x+A). $
For example, $\nabla_{\partial_x}x = 1+Ax$ and on the other hand as a Weyl algebra module $\partial_xx= x\partial_x+1 = xA+1.$