Find output given state space representation
In Laplace space:

$$ sX(s)-x(0)=AX(s)+BU(s) $$

so:

$$ X(s)=(sI-A)^{-1}BU(s)+(sI-A)^{-1}x(0) $$

the second term, due to the initial condition, produces:

$$ x_0(t)=e^{At}~x(0) $$

while the fist term, due to the incoming signal, produces:

$$ x_u(t)=\int_0^te^{A\tau}~B~u(t-\tau)d\tau=\int_0^te^{A(t-\tau)}~~B~u(\tau)~d\tau $$

(where calculation of \$ e^{At} \$ can be seen here )

Finally:

$$ y(t)=Cx(t)=C(x_u(t)+x_0(t)) $$