Question

If x=a(cost+tsint) and y=a(sint-tcost), find d^2y/dx^2

Answer 1

x = a(cost + tsint)
differentiate x with respect to t,
dx/dt = a{d(cost)/dt + d(tsint)/dt]
= a[-sint + {t. d(sint)/dt + sint.dt/dt}]
= a[ -sint + tcost + sint]
= -at.cost
hence, dx/dt = -at.cost -----(1)

y = a(sint - tcost)
differentiate y with respect to t,
dy/dt = a[d(sint)/dt - d(tcost)/dt ]
= a[cost - {t.d(cost)/dt + cost.dt/dt}]
= a[cost +tsint -cost]
= at.sint
hence, dy/dt = at.sint -------(2)

dividing equations (2) by (1),
dy/dx = at.sint/at.cost = tant
now again differentiate with respect to x
d²y/dx² = sec²t. dt/dx ------(3)
now from equation (1),
dx/dt = at.cost
so, dt/dx =1/at.cost put it in equation (3),
e.g., d²y/dx² = sec²t. 1/at.cost
d²y/dx² = sec³t/at

Answer 2

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