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 $x_p(t)=At+B+C\sin t+D\cos t$?

There are different approaches to finding the particular solution to an ODE. The simplest, and I believe the method you are referring to here, is to use the method of Undetermined Coefficients. This method basically amounts to guessing what the solution may be, given the non-homogeneous functions that appear. For an ODE 
$$\sum_{n = 1}^{m} a_n y^{(n)} = g(x)$$
if $g(x)$ contains 
$ae^{bx}$
for some constants $a, b$, then it is logical to guess that the particular solution should contain an exponential of the form $Ae^{bx}$. Similarly, if we have some sines or cosines, we know to get those our particular solution must have a linear combination of sines and cosines. If we have an $n$ degree polynomial, then we should also guess a linear combination of polynomials. 
This method shouldn't be viewed as some sophisticated or strange procedure, it is a very logical form of guessing. Sometimes this procedure will not work or be sufficient and more sophisticated techniques like Variation of Parameters are needed, which expresses the particular solution in terms of the Wronskian and integrals.
