${y}^{\prime}(t)=f(t,y),a\le t\le b,y(a)=\alpha $

apply for the extrapolation technique

we let ${h}_{0}=\frac{h}{2}$ and use Euler method with ${w}_{0}=\alpha $

${w}_{1}={w}_{0}+{h}_{0}f(a,{w}_{0})$

and then midpoint method

${t}_{i-1}=a,{t}_{i}=a+{h}_{0}=a+\frac{h}{2}$ to produce

approximation to $y(a+2{h}_{0})$

${w}_{2}={w}_{0}+2{h}_{0}f(a+{h}_{0},{w}_{1})$

${y}_{1,1}=\frac{1}{2}[{w}_{2}+{w}_{1}+{h}_{0}f(a+2{h}_{0},{w}_{2})]$

then why this form results $O({h}_{0}^{2})$ approximation to $y({t}_{1})$

apply for the extrapolation technique

we let ${h}_{0}=\frac{h}{2}$ and use Euler method with ${w}_{0}=\alpha $

${w}_{1}={w}_{0}+{h}_{0}f(a,{w}_{0})$

and then midpoint method

${t}_{i-1}=a,{t}_{i}=a+{h}_{0}=a+\frac{h}{2}$ to produce

approximation to $y(a+2{h}_{0})$

${w}_{2}={w}_{0}+2{h}_{0}f(a+{h}_{0},{w}_{1})$

${y}_{1,1}=\frac{1}{2}[{w}_{2}+{w}_{1}+{h}_{0}f(a+2{h}_{0},{w}_{2})]$

then why this form results $O({h}_{0}^{2})$ approximation to $y({t}_{1})$