Consider the differential equation:

$\frac{dy}{dx}=-y^2,y(0)=1$

Answer the next questions based $on$ this differential equation. For reference, the exact solution $to$ this

problem $is$ given $byy(x)=\frac{1}{x+1}.$

$(a)(5$ points$)$ Perform $2$ iterations $of$ Euler's method with $h=0\mathrm{.}1.$ What $is$ the resulting value $ofyatx=0\mathrm{.}2?$

$(b)(10$ points$)$ Perform $2$ iteration $of$ Heun's method $(predictor$ corrector$)$ with $h=0\mathrm{.}1.$ What $is$ the resulting

value $ofyatx=0\mathrm{.}2?$

$(c)(5$ points$)$ Perform $1$ iteration $of$ Midpoint method $(predictor$ corrector$)$ with $h=0\mathrm{.}2.$ What $is$ the

resulting value $ofyatx=0\mathrm{.}2?$