Euler to Trigonometric - Particular solution I am back to studying after some years out and I simply forgot the steps that allowed this to function. I recall this is basic, but I can't seem to crack the thing. This is a particular kind of 2nd degree differential equation - one in which the damping is neglected and thus the structure should vibrate endlessly.
The textbook has:
$$
A_1e^{i\lambda t}+A_2e^{-i\lambda t}
$$
Which yields by using Euler's formula the following form:
$$
A_1\sin(\gamma t)+A_2\sin(\gamma t)
$$
I am aware of Euler's formula. What I can't do is making the first equation resemble the exponential => sine equation to consume the exponential and turn them into pure sines.
It is shameful, I know, but any help would be greatly appreciated. 
Thanks!

You have confused $\gamma$ and $\lambda$ and must have a typo (please check closely).  But assuming you fix that:  Use Euler's equation, $e^{i \lambda t} = \cos (\lambda t) + i \sin (\lambda t)$ and substitute, knowing that $\cos (x) = \cos (-x)$ and $\sin (x) = -\sin (-x)$.  
