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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.\(\frac{dt}{dx}\) 

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