How do I find out particular solution for my differential equation? How do I find out particular solution for my differential equation?
For example $\ddot{x}+4\dot{x}+4x=t+1+\sin t$.
Can someone explain me why is particular solution here $x_p(t)=At+B+C\sin t+D\cos t$?

Your equation is linear, so you can decompose its solution into three parts. In particular, let $x = u+v+z$, where
$$
\begin{cases}
\ddot{u} + 4\dot{u} + 4u=1\\
\ddot{v} + 4\dot{v} + 4v=t\\
\ddot{z} + 4\dot{z} + 4z=\sin t
\end{cases}
$$
You can easily verify that if $u,v,z$ solve the above equations, then $x=u+v+z$ is a solution to your equation (thanks to linearity). From here, it is easy to find a particular solution for each of the three equations. 
When looking for a particular solution, you are free to find any solution. A good start is with a guess that is "similar" to the right hand side. Hence, you start with $u=A$, $v=Bt$ and $z=C\sin t + D\cos t$, and you try to see if, for a particular choice of the constants, they actually are solutions. Notice that when the right hand side is a sine or cosine, the solution will in general include both.
That said, there is no universal good guess for particular solutions. There are good guesses for linear equations like yours, but one must pay attention to some details. For instance, you cannot use a guess that is also a solution of the corresponding homogeneous equation. But other than that, the rule of thumb is: if the RHS is a polynomial of degree $k$, try a polynomial of degree $k$, if it is a sine/cosine, try a sum of sine and cosine with the same frequency, if it is an exponential $e^{at}$, try a multiple of such exponential. The good guesses become more complicated if you have more complicated right hand sides, but in your case, you have something relatively simple.
