TITLE: What exactly is steady-state solution?
QUESTION [11 upvotes]: In solving differential equation, one encounters with steady-state solution. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t \rightarrow \infty$. But the steady-state solution is given as $f(t)$, and this means that the solution is a function of $t$ - so what is this $t$ being in limit?

REPLY [2 votes]: A steady state solution is a solution for a differential equation where the value of the solution function either approaches zero or is bounded as t approaches infinity. It sort of feels like a convergent series, that either converges to a value (like f(x) approaching zero as t approaches infinity) or having a radius of convergence (like f(x) being bounded above and below as t goes to infinity).
So if the solution function for your differential equation has an upper and lower bound or asymptotically approaches some constant value as t approaches infinity, then it is a steady-state solution.
A function solution that is not bounded or has no constant asymptotic behavior is not steady-state, and feels like a divergent series.
DISCLAIMER: I'm using the concept of series to give you a feel for how this works, I'm not implying series are relevant in steady-state solutions to differential equations. I mean, they very well might be; but I have no idea about that myself.
Source: I'm a math & stats professor