Let $y''+\lambda y=0$, and $\lambda<0$. Then
$ y(x)=c_1e^{-\sqrt{-\lambda}x}+c_2e^{\sqrt{-\lambda}x}\stackrel{?}{=}c_3\cosh\sqrt{-\lambda}x+c_4\sinh\sqrt{-\lambda}x. $
I do not understand how the RHS of this obtained. I know that use of the equalities
$ \cosh x=\frac{e^x+e^{-x}}{2}\qquad\text{and}\qquad\sinh x=\frac{e^x-e^{-x}}{2} $
is necessary, but the constants $c_i$ make this difficult.