Problem $8.(10$p$)$ Consider the following initial value problem $($IVP$),$

$y^{\text{'}}=1+t^{2}y$

$y(0)=1,$tin$[02]$

$($a$)$ For $h=0\mathrm{.}5$ calculate two steps of Euler's Method, i$.$e$.,$ find $(t_1,w_1)$ and $(t_2,w_2),$ by hand calculation.

$($b$)$ For $h=0\mathrm{.}5$ calculate one step of Heun's Method, i$.$e$.,$ find $(t_1,w_1),$ by hand calculation.

$($c$)$ Check your results for parts $($a$)$ and $($b$)$ using MATLAB. Write down MATLAB commands to obtain $(t_1,w_1)$ and $(t_2,w_2)$ for part $($a$)$ and $(t_1,w_1)$ for part $(b).$