Question

If $x=a(cost+tsint)$ and $y=a(sint-tcost)$, find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

$x=a(\cos t+t\sin t)$ $y=a(\sin t-t\cos t)$

$\frac{dx}{dt}=a(-\sin t+t\cos t+\sin t)$ $\frac{dy}{dt}=a(\cos t+t\sin t-\cos t)$

$\Rightarrow \frac{dx}{dt}=at\cos t$ $\Rightarrow \frac{dy}{dt}=at\sin t$

$\Rightarrow \frac{d^{2}x}{d{t}^2}=at(-\sin t)+a\cos t$ $\Rightarrow \frac{d^{2}y}{d{t}^2}=at\cos t+a\sin t$

$\frac{d^{2}y}{d{x}^2}=\frac{at\cos t+a\sin t}{a\cos t-at\sin t}$

$=\frac{t\cos t+\sin t}{\cos t-t\sin t}$

$=\frac{t+\tan t}{1-t\tan t}$ .