Question

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

Answer 1

Finding,$\frac{dy}{dx}=\frac{dy}{dx}\frac{dt}{dt}$

$=\frac{dy}{dt}$ $=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Calculating $\frac{dy}{dt}$,

$y=b(sint-tcost)$

Diff wrto x,$\frac{dy}{dt}=\frac{d\left(b\right(sint-tcost\left)\right)}{dt}$

$\frac{dy}{dt}=b\frac{d(sint-tcost)}{dt}$

$\frac{dy}{dt}=b(\frac{dsint}{dt}-\frac{tcost}{dt})$

$\frac{dy}{dt}=b(cost-\frac{tcost}{dt})$

$\frac{dy}{dt}=b(cost-(\frac{dt}{dt}cost+\frac{dcost}{dt}t\left)\right)$

$\frac{dy}{dt}=b(cost-(cost+(-sint)t\left)\right)$

$\frac{dy}{dt}=b(cost-cost+t.sint)$

$\frac{dy}{dt}=b(0+t.sint)$

$\frac{dy}{dt}=b.t.sint$

$\frac{d^{2}y}{d{x}^2}=\frac{{sec}^{3}t}{b.t}$