Solving a differential equation with a complex number as a coefficient

I am trying to solve the following differential equation;
\begin{equation}
y'' - iy = 0
\end{equation}
By following the usual method of solving, I get my characteristic equation $\lambda^{2} - i = 0$, which then gives me the general solution of
\begin{equation}
y = cos(\sqrt{i}x) + sin(\sqrt{i}x)
\end{equation}
I want to know if there is any way I can remove the $i$ term from inside the brackets as to get real solutions. I have tried using an expression for $e^{ix}$ but cannot cancel out the $i$ and the $\sqrt{i}$

I also tried this with a similar ODE, as shown below,
\begin{equation}
y'' + iy = 0
\end{equation}
Where I got a characteristic equation of $\lambda^{2} + i = 0$, giving me a general solution of
\begin{equation}
y=e^{i\sqrt{i}x} + e^{-i\sqrt{i}x}
\end{equation}
However, I am still not sure how to cancel this out so that in cos/sin form I have no complex term inside the brackets.

Boundary conditions do not affect the problem at this stage - I'm simply interested in removing the complex term from the brackets.

I hope this is clear enough - any help would be appreciated.