In Example 1.1.4 of the book Grobner Deformations of Hypergeometric Differential Equations, it is stated without proof that

$$\partial^2 \in D\cdot \langle x\partial^4, x^3\partial^2 \rangle ~~~~~~~~~~~~~~~~(\star)$$

where $D$ denotes the Weyl algebra over $\mathbb{k}[x]$, and $ D\cdot \langle x\partial^4, x^3\partial^2 \rangle$ denotes the left-ideal generated by the operators $x\partial^4,x^3\partial^2$. Since Example 1.1.4 is at the very beginning of the book, and no proof is given by the authors, I'm presuming that there a simple way to verify this (i.e. without using the machinery of the main text).

My current line of attack:

So far, this is my thinking: since $D$ is a domain, the equation $(\star)$ is equivalent to solving the following linear equation in the non-commutative ring $D$:

$$1=D_1x\partial^2 +D_2 x^3. ~~~~~~~~~~~~~~~~$$

where $D_1,D_2\in D$ are unknowns. I am able to prove (by brute force calculation) that $\text{ord}\,D_2\geq 1$. Obviously, $\text{ord}\, D_2 = \text{ord}\, D_1+2$. The brute-force calculation in the general case quickly spirals out of control.