Suppose that we want to numerically solve the initial value problem x'(t)=f(x,t), x(0)=x_0. The modified Euler's method $x(t+h)=x(t)+hf(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$
My question, in fact this question from Kincaid and Cheney's book, how to use Richardson extrapolation on Euler's method with step size $h$ and $h/2$ in order to derive the Modified Euler method?
I can improve the method with simply saying, Euler method is x(t+h)=x(t)+hx'(t)+Kh^2 then using method of Richardson extrapolation for general methods but i don't see how to get modified Eulers.
Now these are What I think
Since x'(t)\approx\frac{1}{h}(x(t+h)-x(t)), so say L = x'(t) and $\phi(h) = \frac{1}{h}(x(t+h)-x(t))$ and we have
$L = \phi(h)+hK+O(h^{2})$ and $2L = 2\phi(\frac{h}{2})+2K\frac{h}{2}+O(h^{2})$ then substraction second from the first we have $L= 2\phi(\frac{h}{2})-\phi(h)+O(h^{2})$
Now the question is that how can we show that $2\phi(\frac{h}{2})-\phi(h) = f(t+\frac{1}{2} h,x(t)+\frac{1}{2} hf(t,x(t)))$ Still have no clue,
Thanks for help and hints