Consider $\mathbb{R}$ with trivial derivation, $\mathbb{R}\{x\}$ the ring of differential polynomials in $x,$ and let $J$ be the differential ideal generated by $x''+4x.$ In the quotient $\mathbb{R}\{x\}/J$ we take the ideal generated by $a'^2+4a^2+1,$ where $a$ is the class of $x.$ I want to prove that this I is a differential prime ideal. The fact that $I$ is a differential ideal is straightforward, but I have some problems with proving that $I$ is prime. With definitions, I have not find out anything. Can someone give me an idea? I was wondering if there is some useful property with ring of infinitely many variables but I think that I have to use property the differential properties of these ideals...
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