Would anyone be able to show me how to solve the question in the link? Or give me guidance on where to begin? I'm not sure what is meant by finding the steady-state solution, or how I would go about doing that. (It's a niche problem, but the math should still be pretty general I think.)
On
Your system of ordinary differential equations is autonomous, so steady state solutions are just those corresponding to the values of $(n, \varphi)$ for which both rhs are zero (that is, to the zeros of the vector field). From the second equation we obtain $\varphi = 0$ or $\frac{B n}{\tau} - C = 0$. Plugging $\varphi = 0$ to the first equation we get $n = AI\tau$. Now, $\frac{B n}{\tau} - C = 0$ together with $n = \frac{AI\tau}{1 + B\varphi}$ (obtained from the first equation) gives $\varphi = \frac{AI}{C} - \frac{1}{B}$. Therefore there are two steady states, $$ \begin{cases} n = AI\tau \\ \varphi = 0 \end{cases} \quad \begin{cases} n = \frac{C\tau}{B} \\ \varphi = \frac{AI}{C} - \frac{1}{B} \end{cases} $$ (probably the first one has no physical meaning).
Steady state solution is a solution that does not change over time.
So, if the solution does not depends on time it means that $\frac{d}{dt}=?$