y′(t)=f(t,y), a<=t<=b, y(a)=alpha

${y}^{\prime }\left(t\right)=f\left(t,y\right),a\le t\le b,y\left(a\right)=\alpha$
apply for the extrapolation technique
we let ${h}_{0}=\frac{h}{2}$ and use Euler method with ${w}_{0}=\alpha$
${w}_{1}={w}_{0}+{h}_{0}f\left(a,{w}_{0}\right)$
and then midpoint method
${t}_{i-1}=a,{t}_{i}=a+{h}_{0}=a+\frac{h}{2}$ to produce
approximation to $y\left(a+2{h}_{0}\right)$
${w}_{2}={w}_{0}+2{h}_{0}f\left(a+{h}_{0},{w}_{1}\right)$
${y}_{1,1}=\frac{1}{2}\left[{w}_{2}+{w}_{1}+{h}_{0}f\left(a+2{h}_{0},{w}_{2}\right)\right]$
then why this form results $O\left({h}_{0}^{2}\right)$ approximation to $y\left({t}_{1}\right)$
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Michelle Chavez
This looks more like an order-blackuced RK3 method. Using the standard Runge-Kutta notation, you get
$\begin{array}{rlrl}{k}_{1}& =hf\left(a,{w}_{0}\right),& {w}_{1}& ={w}_{0}+\frac{1}{2}{k}_{1}\\ {k}_{2}& =hf\left(a+\frac{1}{2}h,{w}_{0}+\frac{1}{2}{k}_{1}\right),& {w}_{2}& ={w}_{0}+{k}_{2}\\ {k}_{3}& =hf\left(a+h,{w}_{0}+{k}_{2}\right)\\ {y}_{1,1}& =\frac{1}{2}\left({w}_{1}+{w}_{2}+\frac{1}{2}{k}_{3}\right)\\ & ={w}_{0}+\frac{1}{4}{k}_{1}+\frac{1}{2}{k}_{2}+\frac{1}{4}{k}_{3}\end{array}$
This then gives the Butcher tableau
$\begin{array}{cc}{c}_{1}& \\ {c}_{2}& {a}_{21}\\ {c}_{3}& {a}_{31}& {a}_{32}\\ & {b}_{1}& {b}_{2}& {b}_{3}\end{array}=\begin{array}{cc}0& \\ \frac{1}{2}& \frac{1}{2}\\ 1& 0& 1\\ & \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\end{array}$
where the first and second order conditions
${b}_{1}+{b}_{2}+{b}_{3}=1\phantom{\rule{0ex}{0ex}}{c}_{1}{b}_{1}+{c}_{2}{b}_{2}+{c}_{3}{b}_{3}=\frac{1}{2}$
are both satisfied.
Not exactly what you’re looking for?
Trevor Rush
The solution to the initial value problem has the series
$y\left(a+h\right)=\alpha +f\left(a,\alpha \right)h+\frac{{f}_{t}\left(a,\alpha \right)+f\left(a,\alpha \right){f}_{y}\left(a,\alpha \right)}{2}{h}^{2}+O\left({h}^{3}\right)$
where ${f}_{t}=\mathrm{\partial }f/\mathrm{\partial }t$ and ${f}_{y}=\mathrm{\partial }f/\mathrm{\partial }y$. On the other hand, compute the series for ${y}_{11}$ and you should also find
${y}_{1,1}=\alpha +f\left(a,\alpha \right)h+\frac{{f}_{t}\left(a,\alpha \right)+f\left(a,\alpha \right){f}_{y}\left(a,\alpha \right)}{2}{h}^{2}+O\left({h}^{3}\right)$