Answer
$y(x)=e^{x} -\cos x$
Work Step by Step
Solve the auxiliary equation for the differential equation. $$r^3-r^2+r-1=0$$
Roots are: $(r-1) (r^2+1) =0 \implies r_1=1; r_2=-i; r_3=i$
This implies that there are $\bf{3}$ independent solutions to the differential equation and the general equation is equal to $y(x)=C_1 e^{x} +C_2 \cos x+C_3 \sin x$
Next, after applying the initial conditions we get: $C_1=1; C2=-1; C_3=0$
Therefore, $y(x)=e^{x} -\cos x$ is the solution to the initial-value problem.