Q$.$ pts$)$ Find the Laplace transforms of the following differential equations. Give your answers as $Y(s)=$dots.

a$)3\frac{dy(t)}{dt}+4y(t)=5u(t)$

$IC$: $y(0)=5$;$u(0)=0$;$u(t)=6$ for $t0$

b$)6\frac{d^{2}y(t)}{d{t}^2}+7\frac{dy(t)}{dt}+8y(t)=2u(t)$

$IC$: $y(0)=1$;$y^{\text{'}}(0)=0$;$u(0)=1$;$u(t0)=2$

c$)\frac{d^{2}y(t)}{d{t}^2}+2\frac{dy(t)}{dt}+3y(t)=-4\frac{du(t)}{dt}+5u(t)$

$IC$: $y(0)=0$;$y^{\text{'}}(0)=0$;$u(0)=0,u(t)=5$ for $t0$

d$)8\frac{d^{2}y(t)}{d{t}^2}+7\frac{dy(t)}{dt}+4y(t)=2e^t$

$IC$: $y(0)=0$;$y^{\text{'}}(0)=4$

Q$.2(20$ pts$)$ Find $y(t)$ for the following functions.

a$)Y(s)=\frac{s^2+1}{s^2(s+2)}$

b$)Y(s)=\frac{5s^2+8s-5}{s^2(s^2+4s-5)}$

c$)Y(s)=\frac{3s^2+2s+3}{s^2+3s+2}$

d$)Y(s)=\frac{1}{s}+\frac{2e^{-2s}}{s^2+2s}$

Q$.3,120$ pts$)$ Find the initial and final values of the following functions. Indicate initial and final values clearly.

a$)Y(s)=\frac{5+12s^2-2s^3}{s^2(s-9)(s-1)}$

b$)Y(s)=\frac{s}{s^2+1}+\frac{4+12s+7s^2}{s^2}$

c$)Y(s)=\frac{5s^4+3s+4}{(2s+1{)}^3(s-4{)}^2}$

d$)Y(s)=\frac{9s^4+7s+19}{(8s+7{)}^3(3s-10{)}^2}$