How do we know $\sin$ and $\cos$ are the only solutions to $y'' = -y$? According to Wikipedia, one way of defining the sine and cosine functions is as the solutions to the differential equation $y'' = -y$.
How do we know that sin and cos (and linear combinations of them, to include $y=e^{ix}$) are the only solutions to this equation?
It seems like a clever combination of inverse polynomials or exponentials could do the same.

Suppose $y=y(t)$ is a solution. We can solve this by subtitutions. Let the parameter of the following functions ($x,y,r,\theta$, to be introduced later) be $t$. Consider the substitution $x=y'$. Then this becomes a system of equations:
$$x'=-y,\quad y'=x$$
By another substitution $x=r\cos\theta,\ y=r\sin\theta$ we get
$$r'=0,\quad \theta'=1$$
(I don't have a pen now so maybe this could be $-1$ instead) Solve this you get
$$r=c_1,\ \theta=t+c_2$$
and the original solution is $y=r\sin\theta=c_1\sin(t+c_2)$ which is a linear combination of $\sin$ and $\cos$
