I found this excercise in a book (<http://logosfoundation.org/kursus/music_math.pdf> chapter 4.3):

> 
> Show that the function $f(t) = A \sin(at) + B \sin(bt)$ is periodic when
>  the ratio of $a$ to $b$ is a rational number, and non-periodic if the
>  ratio is irrational. [Hint: Differentiate twice and take linear
>  combinations of the result and the original function to get a single
>  sine wave; use this to get information about possible periods]
> 
> 
> 

I saw general proofs, but I'm interested in using the hint

So, the double derirative:

\begin{array}{l}
f''( t) =-a^{2} A\sin( at) -b^{2} B\sin( bt)
\end{array}

And what now? Do I just multiply $f(t)$ by $a^2$ and add it to $f''(t)$? Is that a logical thing to do (I mean I haven't had to add a function to its derirative before)

\begin{array}{l}
a^{2} f( t) +f''( t) =\left( a^{2} -b^{2}\right) B\sin( bt)
\end{array}

Can I conclude anything from this?