In a letter to Bernoulli, Euler said the following about the ordinary differential equation $y'' + y = 0$:
...both the functions $\sin(x)$ and $\cos(x)$ are solutions to this problem, but so it is $e^{ix}$ and $e^{-ix}$...
Then, I've seen some fonts to go out and say:
Euler could then write $e^{ix} = A\cos(x) + B\sin(x)$ and use the initial conditions on the problem to find out $A = 1$ and $B = i$
But this depends on Euler knowing about uniqueness of solutions to ordinary differential equations, from which will follow that $e^{ix}$ must necessarly be a combination of sine and cosine.
Was there a uniqueness result from the times of Euler and company?