Evaluate the following equations using Laplace transform:

$($a$)$

$f^{\text{'}\text{'}}(x)+5f^{\text{'}}(x)+6f(x)=2e^{-x}$ where $f(0)=0$ and $f^{\text{'}}(0)=0$

$($b$)$

$f^{\text{'}\text{'}}(x)-4f(x)=sin2x$ where $f(0)=1$ and $f^{\text{'}}(0)=-2$

Find the inverse transform of:

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

$[15$ Marks$]$