Let us consider a mass-spring system driven by a external force g(t) at time t. The mass of spring system is m>0, the damping constant is 2b>0, the spring constant is k>0, and the displacement from the equilibrium of the mass-spring system at time t is denoted by y(t). So the motion is governed by

To solve this equation, we use the proportional variation-of-parameters method. Therefore to reach the general solution of (43), we first need the corresponding auxiliary equation

We have three cases for finding the solution of the homogeneous part of equation (43):

Let us begin with case (iii) and presume that

The p-Wronskian can be computed by

where

and

Hence we have

Using formulas (42), we get

and

So, taking integrals of (53) and (54), we find the functions γ1(t) and γ2(t). Lastly, by inserting the functions γ1(t) and γ2(t) into the (48) we get the desired result. Note that similar calculations can be readily done for cases (i) and (ii).

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