A system is described by the following second$-$order linear differential equation

$\frac{d^{2}y}{d{t}^2}+5\frac{dy}{dt}+6y(t)=\frac{dx(t)}{dt}+x(t)$

where $y(0^{-})=2,y^{\text{'}}(0^{-})=1,$ and the input $x(t)=e^{-4t}u(t)$

By taking the Laplace Transform of both sides of the equation above, express $Y(s)$ the Laplace Transform of $y(t)$ in terms of $x(s)$ the Laplace Transform of $x(t)$ and the initial conditions $y(0-)$ and $y^{\text{'}}(0-).(10$ points$)$

Find the transfer function $H(s)=\frac{Y(s)}{x(s)}.$ The transfer function is defined when there are no initial conditions. $(10$ points$)$

Solve the differential equation by finding the inverse Laplace Transform of $Y(s)$ found in part $1$ of this question. $(10$ points$).$