Can we define the time Period of $x\sin (x)$? If yes, then how to find it?
My thoughts: $y=0$ after every $\pi$ so it appears as its time period but can we call it as a time period? Since time period is defined as time taken to complete one oscillation. But I'm confused in this case to define its oscillation or a complete cycle. I don't think it's an oscillatory motion because there is no To-fro motion.
Can we define time period for non-oscillatory motion?
Please help me.
In mathematics, a function $f\colon\Bbb R \to \Bbb R$ is called periodic if there exists $\tau\in(0,\infty)$ such that $$ f(t+\tau)=f(t) \quad \text{for all $t\in\Bbb R$.} \tag{1}$$ The minimal $\tau$ with this property is called the period of $f$. Note that this definition does not use terms like "oscillation" or "complete cycle" explicitly at all.
You can check that $f(t)=t\cdot\sin t$ does not fulfill $(1)$ for any $\tau$, so it is not periodic and hence it has no period. You can check this by proving the more general fact that a continuous periodic function on $\Bbb R$ must be bounded, which is obviously not the case here.
By induction, the property $(1)$ yields $$ f(t+k\tau)=f(t) \quad \text{for all $t\in\Bbb R$ and $k\in\Bbb Z$.}$$ In your example, you have $$ f(0+k\pi)=f(0) \quad \text{for all $k\in\Bbb Z$,}$$ which is why you seem to think that the function should be periodic in some sense. Note however that having this property for a certain value of $t$ ($t=0$ in this case) is not the same as having it for all $t$ (and hence being periodic).