QUESTION $1$: EAC$2,$ C$4,$ CO$2,$ WK$1-4,$ WP$1,3,7$

$($a$)$ Use the second order Runge$-$Kutta method $($Ralston$\text{'}$s method$)$ to numerically integrate $y^{\text{'}}-y=-x^2,y(0)=0$ from $x=0$ to $x=2$ with a step size of $0\mathrm{.}5.$

Given a Ralston's value as follows;

$y_{i+1}=y_i+[\frac{1}{3}k1+\frac{2}{3}k2]h$

$k1=f(,yi)$

$k2=f(+\frac{3}{4}h,yi+\frac{3}{4}k1h)$

$(5$ Marks$)$