Consider the initial value problem $y^{\text{'}}=\frac{1}{4}(1+y^2),y(0)=1.$

$(a)$ Apply Euler's method with step size $0\mathrm{.}25on$ the interval $0,\frac{1}{2}to$ approximate $y(\frac{1}{2}).$

$(b)$ Apply Euler's method again, this time with step size $0\mathrm{.}1,$ also $on$ the interval $0,\frac{1}{2}to$ approximate $y(\frac{1}{2}).$