# Obtain the transfer function of the system if y(t)=e^{-t}-2e^{-2t}+e^{-3t} text{ and } x(t)=e^{-0.5t}

Obtain the transfer function of the system if
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Step 1
Laplace transform of ${e}^{-at}$ is $\frac{1}{\left(s+a\right)}$ Thus, Laplace transform

$Y\left(s\right)=\frac{1}{\left(s+1\right)}-2\frac{1}{\left(s+2\right)}+\frac{1}{\left(s+3\right)}$
$=\frac{\left(s+2\right)\left(s+3\right)-2\left(s+1\right)\left(s+3\right)+\left(s+1\right)\left(s+2\right)}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$
$=\frac{\left(6-6+2\right)}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$
$=\frac{2}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$
Step 2
Also, Laplace transform of
$X\left(s\right)=\frac{1}{\left(s+0.5\right)}$
Step 3
Thus transfer function is: $\frac{\left(Y\left(s\right)\right)}{\left(X\left(s\right)\right)}=\frac{\frac{2}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}}{\frac{1}{\left(s+0.5\right)}}$
$=\frac{2s+1}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$