Consider a continuous time LTI system for which the input $x(t)$ and output $y(t)$ are related by the differential equation\\
$\frac{d^2y}{dt^2} + 7\frac{dy}{dt} + 10y(t) = -4x(t)$\
Let $X(s)$ and $Y(s)$ be the Laplace transform of the input and output respectively and $H(s) = \frac{Y(s)}{X(s)}$ be the transfer function.\
• Determine $H(s)$ and show the pole zero pattern along with region of convergence under the causality assumption.\
• Determine the impulse response of the system.\
• Determine the natural response and forced response of the system for $x(t) = e^{-t}u(t)$ and initial conditions $y(0) = -1$ and $\frac{dy}{dt}|_{t=0} = 5$, where $u(t)$ unit step.
