How to analytically solve this ODEs?

Question

I don't think these ODEs have been explicitly solved before, and I'm wondering if anyone can point me to some papers which might help me start.

Here $n$ is an integer and $S_A,S_B$ can be seen as given input.

\begin{align*} \frac{d}{dt}X_A &= -X_A - X_Aw_{AB}X^n_B+S_A,\\ \frac{d}{dt}X_B &= -X_B - X_Bw_{BA}X^n_A+S_B. \end{align*}

Answer 1

(I doubt this system in general is solvable) but a starting point is to consider $S_A = S_B = 0$ and $W_{AB}, W_{BA}$ are constant

$$ \frac{dX_A}{dt} = -X_A(1+W_{AB}X_B^n) \\ \frac{dX_B}{dt} = -X_B(1+W_{BA}X_A^n) $$

so it follows that

$$ \frac{dX_A}{dX_B} = \frac{X_A}{1+W_{BA}X_A^n } \frac{1+W_{AB}X_B^n}{X_B} $$

Now you have a solution possible. Simply solve the differential equation

$$ \frac{dy}{dx} = \frac{y}{1+W_{BA}y^n} $$

Then $$y\left( \int \frac{1+W_{AB}X_B^n}{X_B} dX_B\right) $$ solves your original equation via chain rule.

We can stretch this further by consider non constant $W$ (as long as $W_{BA}$ depends on $X_A$ and $W_{AB}$ depends on $X_B$ this is really fine). We can also let $S_A, S_B$ be finite degree polynomials say of maximum degree $k$, since we can just differentiate the expressions $k$ times to send them to 0 and that general abstract trick above is still workable as you'll now have separable equations.