If `x=a(cost+tsint)` and `y=a(sint-tcost)` , then find the value of `(d^2y)/(dx^2)` .

Explanation

`x=a(cost+t sint) and y=a(sint-t cos t)`

first we will find` frac{d^{2} x}{d t^{2}} and frac{d^{2} y}{d t^{2}}`

and `frac{d^{2} y}{d x^{2}}=frac{frac{d^{2} y}{d t^{2}}}{frac{d^{2} x}{d t^{2}}}`
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x=a(cos t+t sin t) Rightarrow frac{d x}{d t}=-a sin t+a sin t+a t cos t`
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frac{d^{2} x}{d t^{2}}=a cos t-a t sin t`
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y=a(sin t-t cos t) Rightarrow frac{d y}{d t}=a cos t-a cos t+a t sin t`
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frac{d^2 y}{{dt}}={a} sin {t}+{at} cos t`
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therefore frac{d^{2} y}{d x^{2}}=frac{(sin t+t cos t)}{(cos t-t sin t)}`