Answer
$=3+4\sqrt 3 e^{2t}\sin (\sqrt 3 t)$
Work Step by Step
Taking the Laplace transform of both sides of the given differential equation and imposing the initial conditions yields:
$L[x(t)]=\frac{3}{s}+L[\cos t * x(t)]\\
=\frac{3}{s}+L[t].L[x(t)]\\
=\frac{3}{s}+\frac{L[x(t)]}{s^2+1}$
then $L[x(t)]=\frac{3s+3}{s^3-4s^2+s}=\frac{3}{s}+\frac{12}{s^2-4s+1}$
The general solution to the given equation is:
$x(t)=3+2\sqrt 3[e^{(2+\sqrt 3)t}-e^{(2-\sqrt 3)t}]\\
=3+4\sqrt 3 e^{2t}\sin (\sqrt 3 t)$
