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I have the following differential equation and I need to obtain the transfer function of Z / P but there are constants so I cannot factor to obtain the relationship, how could I obtain the transfer function of Z / P?

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    $\begingroup$ Please use the MathJax markup language to write the equations instead of using a photo; right now, it's not clear what you've erased and what you intend to remain. In addition, please clarify which parameters are variable and which are constants. $\endgroup$ May 17 '21 at 23:01
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Consider it as a multi-input, single output system. The inputs are $P$, $P_a$ and $g$, the output is $z$. Whether these inputs are constant over time doesnt matter that much. The laplace transform of this equation then becomes: $$Ms^2Z(s) = AP(s) - AP_a(s) - MG(s)$$ where $P_a(s) = \frac{P_a}{s}$ and $G(s) = \frac{g}{s}$. This rewrites to $$Z(s) = \frac{A}{Ms^2}P(s) - \frac{A}{Ms^3}P_a - \frac{1}{s^3}g$$ What this means is that the dynamics of $Z(s)$ are influenced by 3 parameters. However, as this is a linear equation, the dynamics do not influence each other. As such, the effect of $P(s)$ to $Z(s)$ can be described as: $$\frac{Z(s)}{P(s)} = \frac{A}{Ms^2}$$ Constant factors in a differential equation are usually considered as disturbances in the Transfer function. The influence of these disturbances on the output can be computed the same way (just pick out the part that is multiplied to the factor).

Alternatively, you could also define a new input which is equal to $U(s) = P(s) - \frac{1}{s}P_a - \frac{M}{As}g$ and use that to define the transfer function. This allows you to capture the dynamics in just a single transfer function, but you should remember that only a part of this input is controllable.

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    $\begingroup$ wouldn't the LT be $Ms^2Z(s)=AP(s)-\frac{AP_a-Mg}{s}$ ? $\endgroup$
    – Pete W
    May 18 '21 at 13:26
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    $\begingroup$ @Pete Indeed! I tagged them as inputs without computing the laplace transform of these inputs. I fixed the answer. Thanks :D $\endgroup$
    – Petrus1904
    May 18 '21 at 13:39

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