A certain LTIC system is described by the following differential equation: [ frac{d^{2} y(t)}{d t^{2}}+3 frac{d y(t)}{d

Question:

A certain LTIC system is described by the following differential equation:

\[
\frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=\frac{d x(t)}{d t}+4 x(t)
\]

where $x(t)=e^{-3 t} u(t)$. The initial conditions are $y\left(0^{-}ight)=2$ and $\dot{y}\left(0^{-}ight)=1$. Determine

(a) The characteristic polynomial

(b) The characteristic equation

(c) The eigen values

(d) The zero input response.

(e) The zero state response.

(f) Total response. Use Laplace transform method.