Hallo,
a)
x=F(s)
x'= -x(0) +s F(s)
x''=-s x(0)-x'(0)+s^2 F(s)
---->
2 s^2 F(s) +7s F(s) +3 F(s)= LT{F(t)}
F(s) (2 s^2 +7s +3)= LT{F(t)}
F(s)= LT{F(t)} /(2 s^2 +7s +3)
b)
F(s)=1 /(2 s^2 +7s +3) *1/s
F(s) = 1/( s (2s+1)(s+3))
Ansatz Partialbruchzerlegung:
F(s)=1/( s (2s+1)(s+3))=A/s +B/(s+3) +C/(2s+1)
F(s)= 1/(3s) +1/(25(s+3)) -4/(5(2s+1))
\( x(t)=-\frac{2}{5} e^{-t / 2}+\frac{e^{-3 t}}{15}+\frac{1}{3} \)