Harmonic oscillator in Heisenberg picture

[dyn]

For the harmonic oscillator in 1-D we get the 2nd time derivative of the x Heisenberg operator = -ω² x. When that is integrated it gives x_H (t) = Acos(ω t) +Bsin (ω t) where A and B are time independent operators. My question is why are the constants A and B incorporated into the terms as a multiplicative factor instead of being additive constants ? And what would happen if the term to be integrated already had an operator in it. you wouldn't know whether to place the multiplicative constant before or after the other operator ?

 

[unscientific]

$$a = \ddot x = -\omega^2 x$$

These constants are multiplied because when you sub it back in it should give LHS = RHS.