Solve for an Undetermined Coefficent by using annihilator method. The problem is 
I know that we need to find  and 
There are three roots and at least two of them are complex. 

Edit: Sorry I forgot the D = 1
The  is supposed to  be . The + right next to the sinx is just a typo. 
Now for the  . Since there is a  on the right hand side of the equation, we need to use . Do I expand the equation to take three derivatives? I know that there's going to be a lot of product rule executions. Is there an easier way to do this problem? I've heard about the annihilator method, but I don't know how to use it. I could do variation of parameters, but it produces too many overwhelming variables. 
This is what I got so far for the first derivative of Yp. I figure if I put the a and b aside for a bit and take the product rule for the terms, it may be easier. 
This is a screenshot of my first derivative and my second derivative of Yp

 A: The equivalent system is:
$$y''' - 3 y'' + 4 y' -2y = e^x \cos x$$
The homogeneous solution gives us $m^3 - 3m^2 + 4 m -2 = 0 \rightarrow m= 1, 1 \pm i$, so:
$$y_h(x) = e^x(c_1 + c_2 \cos x + c_3 \sin x)$$
Now, we notice that we have an $e^x \cos x$ term in the particular solution, we multiply by an $x$ term, so we choose:
$$y_p(x) = x~e^x(a \cos x + b \sin x)$$
We now form $y''' - 3 y'' + 4 y' -2y = e^x \cos x$ and solve for the constants.
Yes, you need to form the first, second and third derivative, it is a bit of algebra, but there are also many cancellations. 
Update


*

*$y'_p = e^x x (b \cos x - a \sin x) + e^x (a \cos x + b \sin x) + 
 e^x x (a \cos x + b \sin x)$

*$y''_p = 2 e^x (b \cos x - a \sin x) + 2 e^x x (b \cos x - a \sin x) + 
 e^x x (-a \cos x - b \sin x) + 2 e^x (a \cos x + b \sin x) + 
 e^x x (a \cos x + b \sin x)$

*$y'''_p = 6 e^x (b \cos x - a \sin x) + 3 e^x x (b \cos x - a \sin x) + 
 e^x x (-b \cos x + a \sin x) + 3 e^x (-a \cos x - b \sin x) + 
 3 e^x x (-a \cos x - b \sin x) + 3 e^x (a \cos x + b \sin x) + 
 e^x x (a \cos x + b \sin x)$

*Now, you have to form $y_p''' - 3 y_p'' + 4 y_p' -2y_p$


Spoiler

 $$y(x) =  e^x (c_1 + c_2 \cos x + c_3 \sin x -\dfrac{1}{2} x~ \cos x)$$

Notes: 


*

*$(1)$ Variation of parameters may actually be easier to use.

*$(2)$ You can see some examples of the Annihilator Method

*$(3)$ If you want an alternate method to undertermined coeeficients or the Ann Method, see example $2$ of AN ALTERNATIVE METHOD FOR THE UNDETERMINED COEFFICIENTS AND THE ANNIHILATOR METHODS
