I came across this problem: which of the following statements are true regarding differentiability.

Is the following statement true?If $f$ is twice continuously differentiable in $(a,b)$ and if for all $x\in(a,b)$ , $$f''(x)+2f'(x)+3f(x)=0$$, then $f$ is infinitely differentiable in $(a,b)$.

I understand the argument using induction. However, I am wondering if the following argument makes sense or not?

**My argument:** Solve the differential equation $y''+2y'+3y=0$, we get the general solution $$y=C_1 e^{-x}\sin{\sqrt{2}x}+C_2e^{-x}\cos{\sqrt{2}x}$$
, which is infinitely differentiable.

My friend thinks that my argument is not correct, since I cannot guarantee that all possible $f$ has to be in form of the general solution. I am confused. Is my reasoning correct?