On the slide 8, the curve x1(t) isn't unique. For example, we can find another curve x2(t) which satisifies that x2(0)=x'2(0)=...=x(n-1)2(0)=0 and x2(T)=x'2(T)=...=x(n-1)2(T)=0. Then any curve has such form as x1(t)+a*x2(t) will be a solution.