I'm reading this section from A Theoretical Introduction of Numerical Analysis (Ryaben'kii, Tsynkov) on ODEs and the author states that the scheme below has first-order accuracy, which I think I understand (we use the Taylor expansion). The author then states that one can change the initial condition $u_1$ to achieve a higher order accuracy. How?
Edit: I'm thinking I could try $u_1= u(0) + u'(0)$... but then again I don't think that makes sense. I am stuck.
$$\ {u'(x)+Au(x)|}_{x=0} = {1+x^2|}_{x=0}$$ $$\ {u'(0)} = 1-Ab $$
As you mentioned $$ u_1 = u(0+h) = u(0) + hu'(0) + \frac{h^2}{2}u''(0) ..... $$ $$ u_1 = u(h) = b + h(1-Ab) + \frac{h^2}{2}u''(\epsilon_0)$$
Initial condition in 9.26 $$ u_1 = b + h(1-Ab) $$
$$L_h[u]_h = f^{(h)}+ \delta f^{(h)} $$ 2nd condition can be expressed as below (include other two terms compare it with modified 9.26) $$L_h[u]_h = b + h(1-Ab) + \frac{h^2}{2}u''(\epsilon_0) $$ Order of accuracy is improved from $h$ to $h^2 $