Question

If $x=a(t-\sin t),y=a(1+\cos t)$ then find $\frac{d^{2}y}{d{x}^2}$

Answer 1

Consider the following differential eq.

$x=a(t-\sin t),y=a(1+\cos t)$

Differentiate w.r.t "t" we get..

$\frac{dx}{dt}=a(1-\cos t).....\left(1\right)$

$\frac{dy}{dt}=a\left(\sin t\right)..............\left(2\right)$

Divide eq. 2 iin eq. 1 we get.

$\frac{dy}{dx}=\frac{\sin t}{(1-\cos t)}$

$\frac{d^{2}y}{d{x}^2}=\frac{(1-\cos t)\cos t-{\sin }^{2}t}{{(1-\cos t)}^2}$

$=\frac{\cos t-1}{{(\cos t-1)}^2}$

$=\frac{1}{\cos t-1}$

Hence, this is the required answer.