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

Explanation

`x=a(cos t+t sin t)` and `y=a(sin t-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}}}`


`x=a(cos t+t sin t) Rightarrow frac{d x}{d t}=-a sin t+a sin t+a t cos t`


`frac{d^{2} x}{d t^{2}}=a cos t-a t sin t`


`y=a(sin t-t cos t) Rightarrow frac{d y}{d t}=a cos t-a cos t+a t sin t`


`frac{d^{2} y}{d t^{2}}=a sin t+a t cos t`


`therefore frac{d^{2} y}{d x^{2}}=frac{(sin t+t cos t)}{(cos t-t sin t)}`