Euler's Explained: Solving Homework Equations

[ee1215]

Homework Statement

Could someone explain how they came to this conclusion?
http://hepweb.ucsd.edu/ph110b/110b_notes/node34.html

Homework Equations

I know how to get:
ω1 dot + Ωω2 = 0
ω2 dot - Ωω1 = 0

The Attempt at a Solution

I know euler is e^(i * t) = cos(t) + sin(t)*i but not sure how the notes take it to :
ω1(t) = Acos(Ωt) , ω2(t) = Asin(Ωt)

 

[vela]

They solved the coupled differential equations.