Question $5[10$ marks$]$

Given the initial$-$value problem

$\frac{dy}{dx}=y$;$,y(0)=1$

$($a$)$ Make use of the predictor$-$corrector method to solve the equation numerically for $x=1,$ using $h=\frac{1}{2}.$

$($b$)$ Compare your result with the exact solution.