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transfer function and unit setp response

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(See attached file for full problem description)
For the following RC circuit,
A.) Using the knowledge that the Laplace transform of dX/dt is sX(s) and:
dVout / dt = 1/RC (Vin - Vout)
derive the Laplace transform of the above system, Vout(s)/Vin(s).

B.) Using the knowledge that the Laplace transform for C is 1/(sC) and that the Laplace Transform of R is simply R, re-derive the transfer function for the above circuit, Vout(s)/Vin(s), with the differential equation found as the starting point. (The answer should be the same as A)

C.) Given the results of 12, and the knowledge that the Laplace transform of a unit step is 1/s, calculate the Laplace transform Vout(s) for Vin=unit step.

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By applying Laplace transform, it shows how to find the transfer function and unit setp response of the RC circuit. The solution is detailed and has a '5/5' rating.

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Please see the attached file.

A.) Using the knowledge that the Laplace transform of dX/dt is sX(s) and:
dVout / dt = 1/RC (Vin - Vout)
derive the Laplace transform of the above system, Vout(s)/Vin(s).

...

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