System of ODE's of rational form

Question

I am faced with a system of differential equations of the form

$$ \begin{align} f'(x) &= \frac{\sum_i{p_i(x,f(x),g(x))\mathrm{e}^{i f(x)}}}{\sum_j{q_j(x,f(x),g(x))\mathrm{e}^{j f(x)}}} \\\ g'(x) &= \frac{\sum_i{r_i(x,f(x),g(x))\mathrm{e}^{i f(x)}}}{\sum_j{s_j(x,f(x),g(x))\mathrm{e}^{j f(x)}}} \end{align}, $$ where $p_i, q_j, r_i, s_j$ are (rather unwieldy) polynomials in three variables and the sums have finitely many terms. I point out that the right sides d0n't contain any derivatives.

I can solve the system numerically using a set of initial conditions for $f(0), g(0)$.

Is there a general theory about solving ODE's of this form symbolically?

Answer 1

Not that I know of. Even for the much simpler system, related to Hilbert's $16^{\text{th}}$ problem, $$ f'=P(f,g),\quad g'=Q(f,g),\quad \text{$P$ and $Q$ polynomials,} $$ I do not know of any such theory.