Let $y(t)$ be the solution to $y^{}=9t{e}^{-y}$ satisfying $y(0)=2.$

$($a$)$ Use Euler's Method with time step $h=0\mathrm{.}2$ to approximate $y(0\mathrm{.}2),y(0\mathrm{.}4),$dots,$y(1\mathrm{.}0).$

$\backslash$table$[[k,t_k,y_k$

$($b$)$ Use separation of variables to find $y(t)$ exactly.

$y(t)=ln(\frac{9}{2}{t}^2+e^2)$

$($c$)$ Compute the error in the approximations to $y(0\mathrm{.}2),y(0\mathrm{.}6),$ and $y(1).$

$|y(0\mathrm{.}2)-y_1|=$

$|y(0\mathrm{.}6)-y_3|=$

$|y(1)-y_5|=$