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Application 3.1
This protocol is extracted from research article:
Modeling of pressure–volume controlled artificial respiration with local derivatives
Adv Differ Equ, Jan 14, 2021;

Procedure

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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