$=\sqrt{2}\left[\frac{1}{\sqrt{2}} \sin \omega t-\frac{1}{\sqrt{2}} \cos \omega t\right]=\sqrt{2} \sin \left(\omega t-\frac{\pi}{4}\right)$

It represents a $SHM$ with time period, $T=\frac{2 \pi}{\omega}$

$y=\sin ^{3} \omega t=\frac{1}{4}[3 \sin \omega t-\sin 3 \omega t]$

It represents a periodic motion with time period

$T=\frac{2 \pi}{\omega}$ but not $SHM$

$y =5 \cos \left(\frac{3 \pi}{4}-3 \omega t\right) $

$=5 \cos \left(3 \omega t-\frac{3 \pi}{4}\right) \quad[\because \quad \cos (-\theta)=\cos \theta]$

It represents a $SHM$ with time period, $T=\frac{2 \pi}{3 \omega}$

$y=1+\omega t+\omega^{2} t^{2}$

It represents a non-periodic motion. Also it is not physically acceptable as $y \rightarrow \infty$ as $t \rightarrow \infty$