Find a smooth piece$-$wise quadratic function y $=$ S$($x$)$ defined on the interval x in $[$x$\_0,$ x$\_$n$]$ where x$\_$i $=$ x$\_0+$ ih that approximates the solution of the initial value problem y$=$ f$($x$,$ y$),$ y$($x$\_0)=$ y$\_0$ i$.$e$.,$ S$($xi$)=$ f$($x$\_$i$,$ S$($x$\_$i$))$ and S$($x$\_0)=$ y$\_0.$ Develop the method theoretically and show that it is equivalent to the implicit trapezoid method. Do not use AI or I will report the answer and please use pen and paper. Thank you