Theorem:
If $\{ \phi_1, \phi_2\}$ is a fundamental set of solutions of $$x''+p(t)x'+q(t)x=0,$$ then for any initial values $x(t_0)=x_0, x'(t_0)=y_0$, there are constants $c_1$ and $c_2$ so that $c_1 \phi_1 + c_2 \phi_2$ solves the IVP.
I know that there is a standard way to prove this theorem, see also How do we deduce that $c_1 \phi_1(x)+c_2 \phi_2(x)$ is a solution of the specific initial value problem?
However, my teacher puts down something like this:
Proof:
It is trivial since the IVP has at most one solution.
I don't see this coming! Maybe it is an easy question but I would appreciate if someone can explain this for me.
