My notes state that the method is constructed based on the idea: yk+1=yk+∫f(x,y)dx where the integral is taken from xk to xk+1 We can estimate the integral by considering ∫f(x)dx (from xk to xk+1) =c0fk+c1fk-1 To simplify the equation, we move xk to the origin such that ∫f(x)dx (from 0 to h) =c0f(0)+c1f(-h) Starting from below, I start to get confused. It says "Replace f(x) with the polynomials, we have" f(x)=1 : h=c0(1)+c1(1) f(x)=x : h2/2=c0(0)+c1(-h) Solving for c0 and c1, c0=3h/2 and c1=-h/2 giving yk+1=yk+h/2*(3fk-fk-1) + O(h3) I've gone through the working and understand where the numbers come from, but I have no idea why they replace f(x) with 1 and x. Why not x2 or x3? I've gone through some books but they derive this with the Taylor's series and I also understand that. I just don't understand the part I mentioned.