= Solution
Write $F=\mathcal Lf$. Using the <Laplace transform of a derivative> and the initial data gives
$$
\mathcal L(f^{(4)})=s^4F-1,\qquad\mathcal L(f^{(3)})=s^3F,\qquad\mathcal L(f'')=s^2F.
$$
The transformed equation is $(s^4+2s^3+s^2)F=1$, so $F=1/[s^2(s+1)^2]$. The preceding inverse <Laplace transform> therefore gives
$$
\boxed{f(t)=t-2+(t+2)e^{-t}.}
$$
For a direct check, $f'=1-(t+1)e^{-t}$, $f''=te^{-t}$, $f^{(3)}=(1-t)e^{-t}$, and $f^{(4)}=(t-2)e^{-t}$. These satisfy both the <ordinary differential equation> and all four <initial conditions>.
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