If you consider poles of an integral transform domain to be important to the solution of differential equations: (as usual,) Euler did it first, 1753.
One "importance" of poles is that they're part of a very useful representation for linear systems. They must've appeared when people started looking at functions as built from generating functions, so that'd ...
Distinct poles do not need to share an x or y coordinate. They are classified as distinct as long as they do not share the same x AND y space. So any two poles are either distinct or repeated.
The effect repeated poles have on the impulse response of a filter is a little complicated, but the short answer is that it does change. This brief article describes ...
Marginally stable means that an otherwise stable system has one or more simple poles on the unit circle (in discrete time), or on the imaginary axis (in continuous time).
The consequence of that is that transients don't decay, but they also don't grow without bounds. Marginally stable systems are unstable in the bounded-input bounded-output (BIBO) sense.
Without using equations.
You look at a circuit that involves R and C ( as a simple exmple ).
There is an applied INPUT VOLTAGE SOURCE.
There is an INITIAL VOLTAGE applied to the capacitor.
You are going to SOLVE for a current or a voltage using 2 DRAWINGS. And keep track of both computed values to make the final answer.
[ 1st drawing ] You draw the circuit ...