Does it make sense to think of a non-constant solution to $\frac{dx}{dt}=0$ (A steady state solution)?

Question

For instance if $\displaystyle\frac{dx}{dt}=x-t$, then $\displaystyle\frac{dx}{dt}=0$ at $x=t$

Answer 1

It doesn't really make sense here to say "at $x=t$" because if $x$ always equals $t$, then the equation just says $\frac{dx}{dx}=x-x$, or $1=0$, which doesn't make sense.

The short answer to your question is no. Given the differential equation $\frac{dx}{dt}=0$, we know that the only possible solution is a constant.

There is a rigorous answer which gives us the existence of a certain set of solutions (any constant), and uniqueness of those solutions (there are no others). This page has a rigorous proof of those theorems if you are interested.