Solve the following differential equation:
I know that first I have to find the solution of the homogenous version of this differential equation and then the solution would be $x(t) = x_o(t)+x_p(t)$
where $x_o(t)$ is the solution of the homogenous DE and $x_p(t)$ is a particular solution that you have to 'observe', right? What is that and what is the intuition behind the $x_p$? What I mean is, I need more details about how to solve this kind of DE.