After solving an ordinary differential equation, I got this solution: $$ A\cos(\omega t + \phi) $$ where A and $\omega$ have fixed values (I don't see the relevance to bring them in, thus I won't).
My question is if I'm allowed to consider instead the solution as the following $$ B\cos(\omega (t - t_{0}) + \phi)$$ because that would simplify the calculations (namely, $\phi$ would be $0$).
I guess it is obvious because in that case $\omega t + \phi$ would be the same as $\omega t - \omega t_{0}$ but I still would like to know why I can use this and why this works, from a math point of view (first year though =)).
Well, $$ A\cos(\omega (t - t_{0}) + \phi) = A\cos(\omega t + \Phi) $$ for some constant $\Phi$, namely $\Phi = \phi-\omega t_0$. So if you are allowed $$ A\cos(\omega t + \phi) $$ for all values of the constant $\phi$, then you are also allowed $A\cos(\omega (t - t_{0}) + \phi)$ for all values of constants $t_0$ and $\phi$.