Write a Python algorithm that evaluates

y$($t$)=$ f$($y$,$ t$)$ y$($a$)=$ y$0(3)$

using a first order Taylor series expansion to approximate the derivative and f$($y$,$t$)=$ y $$ t$2+1,$ with y$(0)=0\mathrm{.}5($i$.$e$.,$ a $=0$ and y$0=0\mathrm{.}5).$ Derive the analytical solution and plot your computed results together with the analytical solution for t in $[0,5].$ Comment your code to explain the key steps. Report the time step size $($i$.$e$.,$t$)$ used to achieve a reasonably accurate answer.