Solution
= Solution
The first three initial values vanish and the third derivative at zero equals one. Transforming the differential equation gives
$$
(s^4+4s^3+4s^2)F(s)-1=0,
$$
so $F(s)=1/[s^2(s+2)^2]$. Hence
$$
\boxed{f(t)=\frac14\bigl(t-1+(t+1)e^{-2t}\bigr)}.
$$