# First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation $$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$ where $A_0, A_1, A_2, B$ are some constants, that depend on rates of reactions.

I'm interested if there are any mathematical facts, that known for this type of equation. I doubt that it is possible to find explicit analytical solution in elementary functions. Also I see that this type of equation doesn't admit nonlinear superposition (unlike Riccati equation, for example).

• I guess, we could re-write the scheme of reactions in a linear-algebraic style for initial condition and deal with eigenvectors/eigenvalues: en.wikipedia.org/wiki/Markov_chain (see "Chemistry"-> Michaelis-Menten on that page). Jun 18 '15 at 12:21

Maple classifies this as an Abel equation of type 2A. It can also be considered as a Chini equation with $n=-1$. The Chini invariant depends on $x$ (except in trivial cases: $A_0 = A_2 = 0$), so this does not lead to a closed-form solution. Moreover, Maple's symgen finds no symmetries.
There are some rather special cases with particular solutions. Thus $x = - A_2 y/A_1$ is a solution if $B={\frac {{A_2}\, \left( {A_0}\,{ A_1}+{ A_2} \right) }{{{ A_1}}^{2}}}$