Since you say you "get most of this", I presume you recognized that the characteristic equation was so that and the solution to the associated homogeneous equation is .
You are TOLD in the problem that so that is . That "n" was introduced just to make the writing easier.
The right hand side of the equation is and, because there are no odd derivatives, we try a solution of the form . Then . Putting those into the equation, it becomes
In order for that to be true for all t, we must have
That term can be written as because we have define so
So the "particular solution" is and the general solution is
Our initial conditions are q(0)= 0, q'(0)= 0.
What is the current in the circuit at any time t ?.
What is the charge in the circuit when w = n ?