Question

If $x=a(\cos t+t\sin t)$ and $y=a(\sin t-t\cos t)$, find $\frac{d^{2}y}{d{x}^2}$

Answer 1

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

first we will find $\frac{d^{2}x}{d{t}^2}$ and $\frac{d^{2}y}{d{t}^2}$
and $\frac{d^{2}y}{d{x}^2}=\frac{\frac{d^{2}y}{d{t}^2}}{\frac{d^{2}x}{d{t}^2}}$

$x=a(cost+tsint)\Rightarrow \frac{dx}{dt}=-asint+asint+atcost$

$\frac{d^{2}x}{d{t}^2}=acost-atsint$

$y=a(sint-tcost)\Rightarrow \frac{dy}{dt}=acost-acost+atsint$

$\frac{d^{2}y}{d{t}^2}=asint+atcost$

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