Trouble understanding which constants are arbitrary and need to be eliminated while forming Differential Equation (DE) from its general solution (In contrast to constants which are fine to be left un-eliminated).
Given general solution** and we need to find it's DE$$y=A sin(\omega t + \phi)$$
We then differentiate it twice $$y'=-A\omega cos(\omega t + \phi)$$ $$y''=-A \omega^2sin(\omega t + \phi)$$
and get the final DE as $$y''=-y\omega^2 \qquad....(1)$$
Problems with Final DE [i.e (1)]
- It contains $\omega$ which i considered to be an arbitary constant. This goes against my understanding (see below)
- what is the difference in $\omega$, $A$ and $\phi$ as constants? Why $A$ and $\phi$ were eliminated but not $\omega$?
- According to my understanding, Final DE should have been of order 3 but that turns out to be wrong. Why?
If we have a general solution with say, 2 arbitrary constants then we need to differentiate the general solution 2 times and eliminate ALL of the arbitrary constants to get the DE free of arbitrary Constants.
Is this correct?
- ** It is taken from equation of SHM (y being the displacement).
- But forgetting our knowledge of physics, how can we mathematically justify $\omega$ in (1).