A question on Laplace transform

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SUMMARY

The discussion centers on deriving the Laplace transform ratio Y(s)/X(s) for the relationship y(t) = 1/(x(t) - k). The user contemplates using a Taylor series expansion for the inverse term, which introduces higher-order terms and convolution in the Laplace domain, complicating the analysis. The user also emphasizes the importance of causality in the relationship, suggesting that y(t) should depend on x(t) from k seconds prior. The confusion arises from the lack of specification regarding the variable x in the denominator.

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Debdut
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x(t) and y(t) are related by y(t)=1/(x(t) -k), how should I derive Y(s)/X(s)?
 
I'm thinking of expanding the inverse term in its Taylor series form. But it would involve terms like (x(t))^2, (x(t))^3, etc if I am right. That would lead to convolution in Laplace domain which according to me is becoming more complicated!
 
I cannot make sense of the question. Here is what I think, y is the "output" and x is the "input" and the relationship is supposed to be y(t) = 1 / x( t - k )
Note I have put the " - k " inside the function argument. This way it has y(t) depending on what x(t) was k seconds ago. This makes more sense since input/output signals in the time domain should be causal and not responding instantaneously. Although maybe I'm missing the point of the question entirely.
 
Then again, why do they put x downstairs without even specifying what it is?
 

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