# If x = a(cos t + t sin t) and y = a(sin t - t cos t), 0 < t < π/2,find d^2x/dt^2,d^2y/dt^2 and d^2y/dx^2. ​

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

find $\frac{d^2x}{dt^2},\frac{d^2y}{dt^2}$ and $\frac{d^2y}{dx^2}.$

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Given,

x = a (cos t + t sin t)

Differentiating both sides with respect to t, we get

$\frac{dx}{dt}$ = a (-sin t + t cos t + sin t)

⇒ $\frac{dx}{dt}$ = at cos t  ...(i)

Differentiating again with respect to t, we get

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

= a (cos t + t sin t)

Again,

y = a (sin t - cos t)

Differentiating with respect to t, we get

$\frac{dy}{dt}=$ a (cos t - t sin t - cos t)  ...(ii)

⇒ $\frac{dy}{dt}=$ at sin t

Differentiating again with respect to t, we get

$\frac{d^2y}{dt^2}$  a (t cos t +sin t)

Now,

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

[from (i) and (ii)]

⇒ $\frac{dy}{dx}=$ $\frac{at\,sin\,t}{at\,cos\,t}$

⇒ $\frac{dy}{dx}=$ tan t

Differentiating again with respect to x, we get

$\frac{d^2y}{dx^2}=$ sect.$\frac{dt}{dx}$