Greetings I have a question about the basic existence and uniqueness theorem for nth order homogeneous linear differential equations on a certain interval.If $\displaystyle f(x_0)=0,f'(x_0)=0,...,f^{(n-1)}(x_0) $ where x, then$\displaystyle f(x)=0$ on the whole interval. I did not understand the logic behind this, can someone explain it to me? I would be grateful. I think this is also valid for first order differential equations if $\displaystyle f(x_0)=0$ then $\displaystyle f(x)=0$ if I am not mistaken?

Thanks