It can be shown that . . .

 (3)                                   

The general solution to a second order differential equation is:

(1")                                    

In Equation 1", the homogeneous solution represents the transient response of the system to the introduction of the periodic motion. The particular solution represents the long-term (steady-state) solution. In this case, we only find the particular solution using the method of undetermined coefficients. Guess the solution of the form:

 (2")                                            

Take the first and second derivatives of Equation 2" yields:

(3")      and                               
(4")                                   

Substituting Equation 2", Equation 3" and Equation 4" into Equation 3:

(4")        

Sorting the terms in Equation 4" yields:

(5")          

The cos[[Omega]]t term must equal zero for this equation to be true, and the sin[[Omega]]t coefficients on both sides must equal each other. This leaves:

(6")       and               
(7")                   

Combining Equation 6" and Equation 7" to solve for C and D yields:

(8")        and                       
(9")                                  

We can now substitute Equation 8" and Equation 9" into:

(2")                                             

(10")                                                 

Algebraic manipulation gives the amplitude response as:

(5)                            

The phase response is given by:

       
(11")     

Rearranging Equation 11" gives:

(6)