Answer
$x(t)=-4+2e^{-t}+2e^t$
Work Step by Step
Taking the Laplace transform of both sides of the given differential equation and imposing the initial conditions yields:
$L[x(t)]=2.\frac{2}{s^3}+L[t * x(t)]\\
=\frac{4}{s^3}+L[t].L[x(t)]\\
=\frac{4}{s^3}+\frac{L[x(t)]}{s^2}$
then $L[x(t)]=\frac{4}{s(s^2-1)}=\frac{4}{s(s-1)(s+1)}=-\frac{4}{s}+\frac{2}{s+1}+\frac{2}{s-1}$
The general solution to the given equation is:
$x(t)=-4+2e^{-t}+2e^t$
