Closed-form solution for this system of ODEs

Question

I am trying to solve the following system (derived from a Michaelis-Menten kinetics model for an enzymatic chemical reaction):

$$\dot{y}_a = r_p x_a - \lambda_p y_a$$

$$\dot{x}_b = \frac{\alpha_0 + \alpha_1 (\frac{y_a}{K})^n}{ 1 + (\frac{y_a}{K})^n} - \lambda_m x_b$$

Ideally, for all $n\in\mathbb Z$, but I would already be quite happy with $n \in \{-2, -1, 1, 2\}$

Currently, I use Fourier series expansions of $x_a$, $y_a$ and $x_b$ to rewrite the system and estimate the values I need...

I am wondering if there might be a closed-form solution to this system?

I think $y_a$ should be rewritable as an exponential function of $x_a$, but reinjecting this in the second equation got me nowhere (even when taking the $\log$... which straightens the fraction, but makes a mess of the rest).

I'd be really grateful for any pointer toward a closed-form solution (or indication that there is none)...

Answer 1

As explained by gorilla in the comments, you consider a system of two differential equations for three unknown functions hence you cannot hope for a solution. An auxiliary result which might help you though is as follows.

If $\dot u=v-cu$ then for every nonnegative $t$, $\displaystyle u(t)=\mathrm{e}^{-ct}\left(u(0)+\int_0^t\mathrm{e}^{cs}v(s)\mathrm{d}s\right).$

If one applies this to your first equation with $u=y_a$, $v=r_px_a$ and $c=\lambda_p$, one gets $y_a(t)$ as a function of $(x_a(s))_{s\le t}$. Likewise for the second equation and $x_b(t)$ as a function of $(y_a(s))_{s\le t}$.