# How to analytically solve this ODEs?

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*}

• Some pointers concerning writing mathematics on Stack Exchange sites: How does one type mathematical formulas on this site? Nov 7, 2022 at 17:28
• there is no closed-form solution, I would just solve them numerically. Nov 7, 2022 at 18:06
• Are $S_A, S_B$ functions or constants and what is $w_{AB}$? Nov 7, 2022 at 18:53
• What does it mean "analytically"? Nov 8, 2022 at 0:38
• @AlexandreEremenko - I think in this context it's the antonym of "numerically" (but you won't find it listed as such in the Thesaurus). Nov 8, 2022 at 2:20

(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.