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