So this is both a question on its own as well as a request for where I can find information on a given topic.
Consider Differential Equations in two variables of the form:
$$P(Z,Z', Z'' ... Z^{[n]}, Y, Y', Y'' ... Y^{[m]}) = 0 $$
Where $P$ is a polynomial of its arguments all of whose coefficients are polynomial functions in a variable x, and $Z$ and $Y$ are polynomials in x as well (the ones we are looking to solve for).
An example
$$y' - z'' = 0$$
Asks for the set of polynomials $y$ and $z$ such that $y = z'$
More involved cases include:
$$y'' + y' = 2z'' + xz' + x^2 z'$$
Which as far as I know doesn't appear to have any trivial solutions.
This is basically the notion of diophantine equations but generalized to functions. How should I go about solving this and what area of mathematics would this correspond to?