Any help appreciated
part i:
$\displaystyle y''+6y'+8y=40$
$\displaystyle p^{2}Y-py(0)-y'(0)+6(pY-y(0))+8Y=\frac{40}{p}$
Using the initial conditons y'(0)=0 and y(0)=0:
$\displaystyle Y(p^{2}+6p+8)=\frac{40}{p}$
$\displaystyle Y=\frac{5}{p+4}-\frac{10}{p+2}+\frac{5}{p}$
Use the inverse Laplace for each and get:
$\displaystyle y=5e^{-4t}-10e^{-2t}+5$