Question

If $x=a(\cos t+t\sin t)$ and $y=a(\sin t-t\cos t),0<t<\frac{\pi }{2}$, find $\frac{d^{2}x}{d{t}^2},\frac{d^{2}y}{d{t}^2}$ and $\frac{d^{2}y}{d{x}^2}.$

Answer 1

$x=a(cost+tsint),y=a(sint-tcost)$

$\frac{dx}{dt}=a(-sint+tcost+sint)=atcost$

$\frac{d^{2}x}{d{t}^2}=\frac{d\left(\frac{dx}{dt}\right)}{dt}=a(cost-tsint)=a(-tsint+cost)$

$\frac{dy}{dt}=a(cost-cost+tsint)=atsint$

$\frac{d^{2}y}{d{t}^2}=a(sint+tcost)$

$\frac{d^{2}y}{d{x}^2}=\frac{d\left(\frac{dy}{dx}\right)}{dx}=\frac{d(\frac{dy}{dt}\times \frac{dt}{dx})}{dt}\times \frac{dt}{dx}$

$=\frac{d\left(\frac{atsint}{atcost}\right)}{dt}\times \frac{1}{atcost}$

$=\frac{d\left(tant\right)}{dt}\times \frac{1}{atcost}$

$=\frac{se{c}^{3}t}{at}$