Is there a numerical solution for a system of three 1st order nonlinear ODE?

Question

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$

$$x' = y $$ $$y'=\sin(x)+z$$ $$z'=y-z$$

I have the following initial conditions;

$$x(0) = 0$$ $$x'(0) = 0 $$

Answer 1

Only to make more obvious what was already obvious, as said in several comments : $$z=y'-\sin(x)$$ $$z'=y''-\cos(x)x'$$ $$z'=y-z=y''-\cos(x)x'=y-(y'-\sin(x))$$ $$y''+y'-y-\cos(x)x'-\sin(x)=0$$ $$x'''+x''-x'-\cos(x)x'-\sin(x)=0$$ Numerical computation of $x(t)$ , $y(t)$ , $z(t)$ requires to state a third condition, for example $x''(0)$ , or $z(0)$, any numerical method used.