coefficient(1): a

integrate (y'') -> -y':
   IC: +1
integrate (-y') -> y
invert (y) -> -y
sin = y

integrate (omega') -> -omega
a * sin -> a_times_sin
omega' = a_times_sin

invert (-omega) -> omega
multiply (omega, omega) -> omega²
multiply (omega², -y) -> -omega²y
y'' = -omega²y