Definition: A differential polynomial is a polynomial with indeterminates $y$, $y'$, $y''$, $\ldots$ with coefficients in $K[x]$, algebra of polynomials with coefficients in a field $K$.
An example of a differential polynomial is $$(1+x)y^2y'+(y'')^2(y''')^5-x^4 yy'y''y'''y^{(4)}-2x. $$ Such polynomials are studied in "Differential Algebraic Geometry" and solution of this polynomials are called "Differential Varieties".
My question: Is it possible to express a differential polynomial as an infinite series with terms of linear differential equations?
For example, is it possible to have an equality of the form $$(y')^2=\sum_{n=1}^{\infty}a_n(x)y^{(n)}? $$ Of course, we need a norm to speak about convergence.
Note that the Taylor series of $(y')^2 $ does not satisfies the condition.
I have an algebra $[E,F]=I, [J,E]=E, [J,F]=-F$, $I$ is central. I would like to have a differential polynomial realization of it, but such that it is not degenerate when $I$ acts as $0$. For example the dif. polynomial $E=i\sqrt{a} x$, $F=i\sqrt{a} \partial_x$, $I=a$, $J=x\partial_x$ where $a$ is a number, is a realization, but $E=F=0$ when $a=0$. Is there another one such that neither $E$ nor $F$ is $0$ when $a=0$ ?