This question is more for general understanding than looking for a specific answer.
I have a theorem that states:
"If Ax=b has a solution x$_p$, then the general solution to the equation is x$_p$ + x$_0$, where x$_0$ ranges over Ker(A)."
I understand that this means the general solution would be x$_p$ + a$_1$x$_1$ + a$_2$x$_2$ + ... + a$_k$x$_k$. However, I keep seeing x$_p$ being referred to as a particular solution. Does particular solution mean that x$_p$ is a unique solution or does it mean that x$_p$ is simply a single solution, possibly amongst an infinite number of solutions? And if it is one of an infinite number of solutions, does it mean we can pick any x$_p$ and add it to x$_n$ where x$_n\in$ Ker(A) to get a general solution?
I hope that makes sense and thank you for any clarification you can provide.
There is a unique solution if and only if the kernel is trivial, i.e., $\ker A=\{0\}$. In this case, $x_p$ is the only solution.
Otherwise, there are many solutions, and you can take $x_p$ to be any specific one of them; your theorem tells how to get all of the other solutions. For each element $x_{ker}\in\ker A$, the quantity $x_p +x_{ker}$ is another solution, and all solutions are of this form (including, incidentally, $x_p$ itself, since it can be written as $x_p+ 0$ and surely $0\in\ker A$).
Note: This assumes we are dealing with a field of scalars, not just a ring in general. The situation is a little different in that case.