Determine the transfer function (H(s)) of the following system using Laplace transform properties. ddot{{y}}+{4}dot{{y}}+{4}{y}=-{x}+{2}dot{{x}} Note: Assume that all initial conditions are zero

Step 1
According to the given information, it is required to determine the transfer function H(s) using Laplace transform properties.$\ddot{y}+4\dot{y}+4y=-x+2\dot{x}$
Step 2
Now represent the above differential equation in transfer function form. It is also given that initial conditions are zero.
$\ddot{y}+4\dot{y}+4y=-x+2\dot{x}$
$L(\ddot{y}+4\dot{y}+4y)=L(-x+2\dot{x})$
$s^{2}Y\left(s\right)-sY\left(0\right)-y^{\prime }\left(0\right)+4[sY\left(s\right)-y\left(0\right)]+4Y\left(s\right)=-X\left(s\right)+2[sX\left(s\right)-x\left(0\right)]$
since initial conditions are all 0:
$s^{2}Y\left(s\right)+4sY\left(s\right)+4Y\left(s\right)=-X\left(s\right)+2sX\left(s\right)$
$(s^2+4s+4)Y\left(s\right)=(-1+2s)X\left(s\right)$
$H\left(s\right)=\frac{Y\left(s\right)}{H\left(s\right)}=\frac{-1+2s}{s^2+4s+4}$