let $u,v,w$ be solutions of $y'''+y=0$ such that $$u(0)=1, u'(0)=0, u''(0)=0$$ $$v(0)=0, v'(0)=1, v''(0)=0$$ $$w(0)=0, w'(0)=0, w''(0)=1$$
show that $u'=-w, v'=u, w'=v$ without findind the solutions.
I suppose all the inequalities are solved in the same way, so one of them should be enough for me to handle the exercise.
if I could find a differential equation such that it has solutions $u'$ and $-w$, then since they share the same initial condition, I could conclude $u'=-w$. but what differential equation would that be?
I don't think I'm going in the right direction and I feel like a hint will be enough for me to solve this.