Mathematics

The two expressions are equivalent because the second is simply a constant multiple (and rearrangement) of the first—if you multiply $y+\tfrac12\cos x=C_2$ by 2, you get $2y+\cos x=2C_2$, and since $C_1$ and $C_2$ are arbitrary constants, $2C_2$ can represent any constant value just as $C_1$ can, so both describe the identical family of solution curves.

The two expressions are equivalent because the second is simply a constant multiple (and rearrangement) of the first—if you multiply $y+\tfrac12\cos x=C_2$ by 2, you get $2y+\cos x=2C_2$, and since $C_1$ and $C_2$ are arbitrary constants, $2C_2$ can represent any constant value just as $C_1$ can, so both describe the identical family of solution curves.